NETWORK ABSTRACTIONS FOR DESIGNING

RELIABLE APPLICATIONS USING WIRELESS SENSOR

NETWORKS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

Vinodkrishnan Kulathumani, B.E., M.S.

* * * * *

The Ohio State University

2008

Dissertation Committee:

Anish Arora, Adviser

Prasun Sinha

Tamal Dey

Paul Sivilotti

Approved by

Adviser

Graduate Program inComputer Science and

Engineering

c© Copyright by

Vinodkrishnan Kulathumani

2008

ABSTRACT

Applications of wireless sensor networks are moving from simply monitoring based

to control based ones and from static network based to pervasive and mobility-centric

ones. But while the applications are rising in scale and complexity, the underlying

network is still resource-constrained and bandwidth limited, prone to contention and

fading. Thus the demands of applications are growing at a faster rate than the re-

sources in the underlying network. This dissertation has addressed the challenge of

reliable application design using wireless sensor networks, by the design and imple-

mentation of network abstractions that bridge the gap between the application and

the network and provide performance guarantees to applications.

The dissertation considers the reliable design of 4 wireless sensor network appli-

cations: (1) distributed pursuer evader tracking with requirement of eventual catch,

(2) distributed pursuer evader tracking with optimal interception, (3) object classi-

fication and track monitoring and (4) distributed control of flexible structures. For

each of these applications, we come up with an appropriate design considering lim-

itations of the underlying network and characterize the network abstractions that

meet application requirements. The network abstractions are then implemented ap-

propriately sometimes using middleware services running in the form distributed /

centralized programs, sometimes by suitably designing the network with the right

density, placement of sensors or sometimes using both.

ii

This dissertation is dedicated to my family

iii

ACKNOWLEDGMENTS

It is a pleasure to thank the many people who made this thesis possible.

Foremost, I would like to thank my Ph.D. advisor, Dr. Anish Arora for giving

me the opportunity to conduct focused research and for sharing with me a lot of his

expertise and research insight. I greatly appreciate his insightful advice on research

and presentation skills.

During the DARPA NEST project, I was fortunate to interact with and learn

from many great scientists and researchers including Dr. Emre Ertin, Dr. Mohamed

Gouda, Dr. Ted Herman, Dr. Sandeep Kulkarni, Dr. William Leal, Dr.Mikhail

Nesterenko, and Dr. Prasun Sinha. Their wisdom has greatly enriched my Ph.D.

experience.

Being a Ph.D. student in an active research university, I have also enjoyed and

appreciated the advice and help from many other professors including Dr. Prasun

Sinha, Dr. Tamal Dey, Dr. Paul Sivilotti, Dr. Randy Moses, Dr. Raj Jain, Dr.

Wu Chi Feng and Dr. Rama Yedavalli. They have made my stay at The Ohio State

University fun and fruitful.

I would like to specially thank Dr. Rajiv Ramnath, who has offered me some

great advice on topics in research, life and career directions.

During my Ph.D. study, I have got the chance to work with many other fellow grad-

uate students, Mukundan Sridharan, Sandip Bapat, Vinayak Naik, Hongwei Zhang,

iv

Santosh Kumar, Murat Demirbas and Lifeng Sang, on shared projects and research

problems. It was a wonderful experience. I would also like to thank my many other

friends for their continued support during my life and study at Ohio State.

I am indebted to my parents Mr. K. K. Mani and Mrs. Lakshmi Mani for

their unconditional love, encouragement and support. My parents have always put

education as a first priority in my life. They taught me to value honesty, courage,

and humility above all other virtues. My parents have always been there for me as

an unwavering support. A special thanks to my brother Ram for always being there

to cheer me up.

Last, but not least, I would like to thank my wife Gayathri. She has been my

source of strength. Her love and support have been enabling me to focus on my

research during the days and nights, the weekdays and weekends.

v

VITA

November 7, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Tenkasi, India

1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B.E. Computer Engineering

June - September 2000 . . . . . . . . . . . . . . . . . . . . . Intern, Microsoft Corporation

2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .M.S. Computer and Information Sci-ence

1999-2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Teaching Associate,The Ohio State University

June - November 2001 . . . . . . . . . . . . . . . . . . . . . . Network Architect, Nayna Networks

March - June 2007 . . . . . . . . . . . . . . . . . . . . . . . . . .Research Intern, Los Alamos NationalLabs

2002-present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Graduate Research Associate,The Ohio State University

PUBLICATIONS

Research Publications

V. Kulathumani and M. Sridharan and R. Ramnath and A. Arora, “Weave: An Ar-

chitecture for Tailoring Urban Sensing Applications across Multiple Sensor Fabrics”,MODUS, International Workshop on Mobile Devices and Urban Sensing, 2008

V. Kulathumani and A. Arora, “Distance Sensitive Snapshots in Wireless SensorNetworks”, International Conference on Principles of Distributed Systems (OPODIS),

2007.

vi

V. Kulathumani, M. Demirbas, A. Arora, and M. Sridharan, “Trail: A DistanceSensitive Network Service for Distributed Object Tracking”, European Conference on

Wireless Sensor Networks (EWSN), 2007

M. Demirbas, A. Arora and V. Kulathumani, “Glance: A Lightweight QueryingService for Wireless Sensor Networks”, International Conference on Principles of

Distributed Systems (OPODIS), 2006

H. Cao, E. Ertin, V. Kulathumani, M. Sridharan and A. Arora, “Differential Gamesin Large Scale Sensor Actuator Networks”, International Conference on Information

Processing in Sensor Networks, IPSN, 2006

E. Ertin, A. Arora, R. Ramnath, M. Nestkerenko, S. Bapat, V. Naik, V. Kulathumani,

M. Sridharan, H. Zhang, H. Cao, “Kansei: A Testbed for Sensing at Scale”, Interna-tional Conference on Information Processing in Sensor Networks, Special Track on

Platform Tools and Design Methods for Network Embedded Sensors (IPSN/SPOTS),2006

V. Kulathumani, P. Shankar, Y. M. Kim, A. Arora, and R. Yedavalli, “Reliable

Control System Design Despite Byzantine Actuators”, Fifth International Conferenceon Multibody Systems, Nonlinear Dynamics and Controls (MSNDC), 2005

S. Bapat, V. Kulathumani, and A. Arora, “Analyzing the Yield of ExScal, a Large

Scale Wireless Sensor Network Experiment”, International Conference on NetworkingProtocols (ICNP), 2005

S. Bapat, V. Kulathumani, and A. Arora, “Reliable Estimation of Influence Fields

for Classification and Tracking in Unreliable Sensor Networks”, IEEE Symposium on

Reliable Distributed Systems (SRDS), 2005

Y. M. Kim, A. Arora, and V. Kulathumani, “On the Effect of Faults in VibrationControl of Fairing Structures”, Fifth International Conference on Multibody Systems,

Nonlinear Dynamics and Controls (MSNDC), 2005

A. Arora, E. Ertin, R. Ramnath, P. Sinha, S. Bapat, V. Naik, V. Kulathumani etal., “ExScal: Elements of an Extreme Scale Wireless Sensor Network”, 11th IEEE

International Conference on Embedded and Real time Computing Systems and Appli-cations, (RTCSA), 2005

vii

M. Demirbas, A. Arora, V. Mittal, and V. Kulathumani, “A Fault-Local Self-StabilizingClustering Service for Wireless Ad Hoc Networks”, IEEE Transactions on Parallel

and Distributed Systems, 2005

A. Durresi, R. Jain, N. Chandhok, R. Jagannathan, S. Seetharaman, and V. Kulathu-mani, “IP over WDM Networks”, Proceedings of IEEE Global Telecommunications

Conference, (Globecom), Volume 4, pp. 2144-2149, 2001

M. Demirbas, A. Arora, V. Mittal, and V. Kulathumani, “A Fault Local Self-StabilizingClustering Service for Wireless Adhoc Networks”, IEEE Transactions on Parallel and

Distributed Systems, Vol. 17, No. 9, pp. 912 - 922, 2006

A. Arora, P. Dutta, S. Bapat, V. Kulathumani, H. Zhang, V. Naik, V.Mittal, H.

Cao, M. Gouda, Y. Choi, T. Herman, S. Kulkarni, U. Arumugam, M. Nesterenko, A.Vora and M. Miyashita, “A Line in the Sand: A Wireless Sensor Network for Target

Detection, Classification and Tracking”, Computer Networks Special edition, Vol. 46,No. 5, pp. 605-634, 2004

V. Kulathumani, N. Chandhok, A. Durresi, R. Jain, R. Jagannathan, and S. Seethara-

man, “Survivability in IP over WDM Networks”, Journal of High Speed Networks,Vol. 10, No. 2, pp. 79-90, 2001

FIELDS OF STUDY

Major Field: Computer Science and Engineering

Studies in:

Networking Prof. Anish AroraDistributed Computing Prof. Prasun Sinha

Prof. Paul SivilottiRandom Signals Analysis Prof. Jose Cruz

viii

TABLE OF CONTENTS

Page

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapters:

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Overview of Approach . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Contributions of the dissertation . . . . . . . . . . . . . . . . . . . 41.2.1 Distributed tracking application with eventual catch . . . . 6

1.2.2 Distributed tracking application with optimal catch . . . . . 71.2.3 Classification of intruders and monitoring their tracks . . . 10

1.2.4 Distributed vibration control . . . . . . . . . . . . . . . . . 14

1.3 Organization of this Thesis . . . . . . . . . . . . . . . . . . . . . . 16

2. Distributed Pursuer Evader Tracking Application with Eventual Catch . 17

2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Sufficient Conditions for Eventual Catch . . . . . . . . . . . . . . . 192.3 Snapshot service for wireless sensor networks . . . . . . . . . . . . 21

2.3.1 Network model and problem statement . . . . . . . . . . . . 242.3.2 Distance sensitive resolution and latency . . . . . . . . . . . 27

2.3.3 Distance sensitive rate . . . . . . . . . . . . . . . . . . . . . 35

ix

2.3.4 Irregular Networks . . . . . . . . . . . . . . . . . . . . . . . 412.4 Using the snapshot service for distributed object tracking . . . . . 47

2.5 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5.1 Pursuer evader games using sensor network . . . . . . . . . 48

2.5.2 Snapshot services . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3. Distributed Pursuer Evader Tracking Application with Optimal Catch . . 53

3.1 Game Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Conditions for optimal interception . . . . . . . . . . . . . . . . . . 56

3.2.1 Optimal pursuit under perfect information . . . . . . . . . . 57

3.2.2 Sampling rate requirements of the optimal pursuit strategy 583.2.3 Effect of Packet Delay . . . . . . . . . . . . . . . . . . . . . 60

3.3 Trail: Network service for distributed tracking . . . . . . . . . . . . 623.3.1 System Model and Specification . . . . . . . . . . . . . . . . 65

3.3.2 Trail on a 2-d Plane . . . . . . . . . . . . . . . . . . . . . . 673.3.3 Implementing Trail in a WSN . . . . . . . . . . . . . . . . . 81

3.3.4 Refinements of Trail . . . . . . . . . . . . . . . . . . . . . . 933.3.5 Modifying Trail Segments . . . . . . . . . . . . . . . . . . . 101

3.3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.3.7 Performance Evaluation . . . . . . . . . . . . . . . . . . . . 110

3.4 Implementation of Trail in a Real Network . . . . . . . . . . . . . . 1163.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 117

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.5 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4. Classification of intruders and track monitoring . . . . . . . . . . . . . . 130

4.1 Influence Field Based Classification and Tracking . . . . . . . . . . 1334.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.1.2 Fault Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.1.3 Estimating the influence field . . . . . . . . . . . . . . . . . 136

4.2 Reliable Estimation Despite Faults . . . . . . . . . . . . . . . . . . 1384.2.1 Tolerating false reports . . . . . . . . . . . . . . . . . . . . 139

4.2.2 Network faults . . . . . . . . . . . . . . . . . . . . . . . . . 1444.3 Putting it all together: Classification and Tracking . . . . . . . . . 154

4.3.1 Composing fault classes . . . . . . . . . . . . . . . . . . . . 1544.3.2 End-to-end reliability . . . . . . . . . . . . . . . . . . . . . 155

4.3.3 Identifying and isolating objects . . . . . . . . . . . . . . . 1564.3.4 Tracking using shape estimation . . . . . . . . . . . . . . . 157

x

4.4 Case study: A Line In The Sand . . . . . . . . . . . . . . . . . . . 1584.4.1 Experimental measurements . . . . . . . . . . . . . . . . . . 159

4.4.2 Determining network density . . . . . . . . . . . . . . . . . 1594.4.3 Effect of network unreliability . . . . . . . . . . . . . . . . . 160

4.4.4 Experimental validation . . . . . . . . . . . . . . . . . . . . 1604.4.5 System performance . . . . . . . . . . . . . . . . . . . . . . 162

4.5 Extensions to the Influence Field Approach . . . . . . . . . . . . . 1624.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5. Distributed Vibration Control . . . . . . . . . . . . . . . . . . . . . . . . 168

5.1 System and Fault Model . . . . . . . . . . . . . . . . . . . . . . . . 1725.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.1.2 Asymptotic Stability Without Faults . . . . . . . . . . . . . 1735.1.3 Fault Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.2 Reliable Control System Design . . . . . . . . . . . . . . . . . . . . 1755.2.1 Reliable Control System Using Redundant Colocated Actuators176

5.2.2 Reliable Control System Without Using Colocated Actuators 1785.3 Application to Beam Vibration Control System . . . . . . . . . . . 182

5.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.1.1 Snapshot services for wireless sensor networks . . . . . . . . 189

6.1.2 Trail: Sensor network service for distributed object tracking 190

6.1.3 Influence field based classification and tracking using wirelesssensor networks . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.1.4 Reliable control system design despite Byzantine actuators . 1916.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Appendices:

A. Proof of Maxima of expression in Eq. 3.11 . . . . . . . . . . . . . . . . . 195

B. Technical Note: On Terminating Points for Tracking Mobile Objects . . . 197

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

xi

LIST OF FIGURES

Figure Page

2.1 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Neighboring level 1 clusters . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Illustrating level 1 tree rooted at j . . . . . . . . . . . . . . . . . . . . . 30

2.4 Summary of results for snapshot algorithms . . . . . . . . . . . . . . 41

2.5 Virtual grid and cells with different densities . . . . . . . . . . . . . . . . 43

2.6 Handling holes in dense networks . . . . . . . . . . . . . . . . . . . . . 45

3.1 The pursuer and evader game . . . . . . . . . . . . . . . . . . . . . . 55

3.2 The P-E trajectory under perfect information . . . . . . . . . . . . . 58

3.3 The P-E trajectory when using the Tsamp update . . . . . . . . . . . 60

3.4 Effect of packet delay . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5 Examples of Trail to an Object P . . . . . . . . . . . . . . . . . . . . . 68

3.6 Updating trailP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.7 Analyzing Trail Stretch Factor . . . . . . . . . . . . . . . . . . . . . . . 73

3.8 Analyzing Trail Stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.9 Effect of b on Update cost . . . . . . . . . . . . . . . . . . . . . . . . 79

xii

3.10 Basic find algorithm in Trail . . . . . . . . . . . . . . . . . . . . . . . . 80

3.11 Find and update algorithm in a WSN grid . . . . . . . . . . . . . . . . . 83

3.12 Trail: State at Node j for Object P . . . . . . . . . . . . . . . . . . . 84

3.13 Trail: Update Actions (U1 and U2) . . . . . . . . . . . . . . . . . . . 86

3.14 Trail: Update Actions (Actions U3 and U4) . . . . . . . . . . . . . . 88

3.15 Trail: Find Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.16 Tolerating failures during find . . . . . . . . . . . . . . . . . . . . . . . 91

3.17 Trail: Additional State at Mote j for Stabilizing Actions . . . . . . . 91

3.18 Trail: Stabilizing Actions . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.19 Level of exploration k = mxq − 2 . . . . . . . . . . . . . . . . . . . . . . 98

3.20 Optimized find: pattern of exploration . . . . . . . . . . . . . . . . . 100

3.21 Optimized find: example . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.22 Find-centric Trail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.23 Trail update costs and Trail stretch factor . . . . . . . . . . . . . . . . . 112

3.24 Trail: find cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.25 Effect of interference on find cost . . . . . . . . . . . . . . . . . . . . . . 114

3.26 Effect of interference on find cost: objects being found collocated . . . . . 115

3.27 Scaling in number of objects (query frequency 0.33 Hz, object update 0.5 Hz)120

3.28 Scaling in number of objects (query frequency 0.5 Hz, object update 0.5 Hz) 120

3.29 Scaling in query frequency (6 objects, object update rate 0.5 Hz) . . . . . 121

3.30 Scaling in object speed (6 objects, query frequency 0.5 Hz) . . . . . . . . 122

xiii

3.31 Trail: analytical comparison . . . . . . . . . . . . . . . . . . . . . . . . 124

4.1 Magnetometer based influence fields for two object types. . . . . . . . 132

4.2 Inversion in a one hop network. . . . . . . . . . . . . . . . . . . . . . 149

4.3 Dealing with inversion using Contention compensation techniques. . . 153

4.4 Impact of network reliability on influence fields in A Line In The Sand 161

4.5 Classification and tracking of a car in A Line In The Sand . . . . . . 162

5.1 Fairing Shaped Payload Installed with Sensors and Actuators . . . . . . . 169

5.2 4-uniform and 7-uniform configurations for the second-degree system . . . 179

5.3 The maximum energy derivative ED(k) in m-uniform configuration system 182

5.4 Energy of the Beam Vibration System . . . . . . . . . . . . . . . . . . . 184

5.5 Energy of the 4-Uniform Configuration Beam Vibration System . . . . . 185

A.1 Finding maxima for f(θ, φ) . . . . . . . . . . . . . . . . . . . . . . . . . 196

xiv

CHAPTER 1

INTRODUCTION

Applications of wireless sensor networks are growing at a faster rate than the re-

sources in the underlying network. In the past ten years, wireless sensor networks

have been typically used for many observation-based applications such as habitat

monitoring and surveillance [2, 4] where the sensors gather a variety of information

and this information is processed centrally or in a distributed manner. But now, ap-

plications are changing from being simply observation based to control based where

the networks perform actuation and control and from being static network based

to mobility-centric ones. The fact that sensor network systems can be really large

scale and can be wireless makes them attractive for industrial and process control

applications such as illumination control [60] and control of distributed parameter

systems [12]. A specific example is the vibration control of a fairing during payload

launch using embedded MEMS components based sensor actuator networks. Increas-

ingly, sensors are being integrated with mobile devices like cell-phones and PDAs to

support applications like on-line patient health monitoring, disaster relief, social net-

working, vehicular networking and mobile gaming and thus wireless sensor networks

are becoming mobile. Additionally the scale at which the networks are operated have

grown from 10s of nodes [2] to several thousands of nodes [4].

1

But while the applications are growing in scale and complexity, the underlying net-

work still consists of resource constrained devices where energy and computational

capabilities are at a premium. The underlying communication can fail in unpre-

dictable ways due to bandwidth limitations, limitations of computational power and

environmental effects. As a result, the network is vulnerable to faults such as infor-

mation loss, delay and data corruption. The challenge in my research has been to

design reliable applications despite the unreliability and the energy constraints in the

underlying network.

In my research, I have dealt with 3 main classes of application namely classifi-

cation, tracking and distributed control. I have considered tracking applications in

two contexts: (1) observation of tracks of mobile objects at a central location, and

a (2) distributed observation and control of tracks of mobile objects. Performance

is critical in most of these sensor network applications. For example, (1) high ac-

curacy and low latency are critical in classification and tracking applications that

are often deployed in military settings to guard perimeters and valuable assets, and

(2) distributed control systems have applications in space missions and nuclear plants

where degradation of system performance may even compromise human safety.

The challenge addressed by this dissertation is the design of sensor networkapplications that scale in a reliable and energy efficient manner despite

the unreliability associated with the underlying network.

1.1 Overview of Approach

In order to address the above challenge, this dissertation proposes specifying the

requirements of the application from the network in terms of suitable network ab-

stractions and then implementing these abstractions. The implementation of the

2

abstractions is performed by either using middleware services that are programs run-

ning in the network or by simply designing the network by choosing the right density

or placement of nodes. In this way, the network abstractions are wrappers around

the network, that conform to certain specifications. The following are two notable

aspects of this strategy.

1. The application layer does not have explicit knowledge of the underlying net-

work parameters and is not involved in tuning of any network parameters. Thus

this is not a cross layer design of the application and the network layer and the

implementation of the application logic is therefore kept simple. The appli-

cation implementation is shielded from the underlying network in the form of

network abstractions.

2. The strategy allows the network abstractions to be implemented in a tunable

way such that depending on application parameters, the network can adjust

itself to meet the specifications and at the same time the overall system is

energy efficient.

The network abstractions can be used to design reliable applications in a couple

of usage models. In the first one, the application and the network layers are de-

signed independently and in a de-coupled manner. The application requirements are

translated to network specifications and either an implementation of this specification

already exists or a network abstraction is implemented on top of the network layer,

that satisfies this specification.

3

However, it may not be always feasible to assume that the specification can be

implemented. This is because, there exists a tension between the application require-

ments and what the network can supply. The application will always demand as much

resources from the network as possible, but meeting those may not be possible. For

example, let us consider the right logic for classification of objects using a wireless

sensor network. Transporting every sensor sample to a centralized base station to

perform the classification is not a sensible idea as it would consume lot of resources.

Moreover, it would cause lot of contention and therefore end up decreasing the over-

all system performance and the quality of the application performance. Therefore, a

co-design of the application and network is needed. The application logic is designed

as per network limitations. Once such an application is designed, the next challenge

is to guarantee that the network meets these requirements and this is done using

network abstractions.

Note that even in the co-design strategy, the application is still shielded from the

low level network details and thus the implementation of the application is therefore

kept simple. This dissertation explores this co-design of application and network using

network abstractions, to design reliable applications using wireless sensor networks.

1.2 Contributions of the dissertation

The dissertation considers the design of the following applications using wireless

sensor networks.

1. Distributed tracking application with eventual catch: In this application, one

or more pursuers are required to eventually catch one or more evaders in a

large region. An underlying sensor network is deployed in the region to detect

4

and track the pursuers and evaders in the network. The tracking application

executes on the pursuer object and uses the sensor network to get the desired

information about the evader objects.

2. Distributed tracking application with optimal catch: Here, we strengthen the

requirements of the tracking application. In this application a large sensor

network is deployed along the perimeter of a valuable asset. Intruders enter

the perimeter with the intention of crossing over to the asset and the objective

of the interceptors is to “catch” the intruders as far away from the asset as

possible. The tracking application executes on the interceptor object and uses

the sensor network to get the desired information about intruder objects.

3. Centralized classification and tracking: The goal of this application is to reliably

classify and tracks objects moving through a region covered by a wireless sensor

network at a central location.

4. Distributed vibration control: A flexible structure such as a beam or a fairing

structure for satellite payload launching is deployed with sensors and actua-

tors. The goal of the application is to detect and control the vibrations in the

structure.

In this dissertation, for each of the above applications, we design an application

logic given the network constraints. We then state requirements from the network

that provide application guarantees and these requirements are translated to network

abstractions. The network abstractions are implemented appropriately sometimes as

middleware services running in the form distributed / centralized programs, some-

times by suitably designing the network with the right density, placement of sensors

5

or sometimes as both. In the following chapters, we formally state the application

model and problem statement, formally specify the network abstractions and provide

details of their implementations. Before that, in the following subsection, we provide

a brief overview of the same.

1.2.1 Distributed tracking application with eventual catch

An underlying sensor network is deployed in the region to detect and track multiple

pursuers and evaders in a region. The tracking application executes on the pursuer

object and uses the sensor network to get the desired information about the evader

objects. We assume that every pursuer object is assigned to a particular evader. The

goal of the tracking application executing on every pursuer object is to eventually

catch the evader that is assigned to that pursuer object.

Application Logic

If the underlying network can provide exact information about object locations

continuously and instantaneously (with no delays) and if the pursuer is faster than

the evader, then an eventual catch is trivial. But getting such perfect information

is infeasible using a wireless sensor network. In our analysis we determine sufficient

conditions on the resolution, latency and rate of information about the evader being

tracked in order to satisfy eventual catch. Specifically, we show that eventual catch

is satisfied if the error in the estimate of distance to the evader decreases linearly

with distance between pursuer and the evader, if the rate at which this informa-

tion is supplied to the evader decreases linearly with distance and if the staleness

in the information supplied decreases linearly with distance, where the constants of

proportionality depend on the relative speeds of the pursuer and the evader.

6

Network abstractions and their implementation

The above application imposes a distance sensitivity requirement on the network

in terms of latency, rate and resolution. We formally define 3 different network

abstractions namely distance sensitive latency, distance sensitive rate and distance

sensitive resolution. In order to implement these abstractions, we design a middleware

service that periodically delivers the global snapshot of the system to all nodes in the

network where the snapshots satisfy the required distance sensitivity properties.

Our service is easily adapted to allow snapshots to be delivered only to a subset of

nodes as opposed to all nodes. They are memory efficient, and are readily realized in

networks with irregular density, networks with arbitrary sized holes, imperfect clus-

tering, and non unit disk radios. We quantify the maximum rate at which information

can be generated at each node so that snapshots are periodically delivered across the

network, the service can of course be operated at lower rates than these. The service

does not require global time synchronization or localization.

Finally, we show our snapshot service is used in the context of the distributed

tracking application and results in satisfying the sufficient conditions for eventual

catch.

1.2.2 Distributed tracking application with optimal catch

In this application a large sensor network is deployed along the perimeter of a

valuable asset. We strengthen the requirement of tracking in this application by

requiring an optimal catch. In other words, intruders enter the perimeter with the

intention of crossing over to the asset and the objective of the interceptors is to

“catch” the intruders as far away from the asset as possible. The interceptors use

7

the underlying wireless sensor network to gather information about intruders and

intercept the intruders in an optimal manner.

Design of Application Logic

It is known that there exist optimal min-max strategies for the intruder interceptor

application when perfect state information is available to the objects. However it

is unreasonable to assume in a constrained sensor network that information about

intruder objects is instantaneously provided to the interceptor objects. Therefore we

study the sampling rate requirements for the interceptor object based on the state of

the system to preserve the optimality of the pursuit strategy.

We prove that latency with which the exact state of the intruder is provided to

the requesting interceptor should decrease proportionally with decreasing distance be-

tween interceptor and intruder in order to to guarantee that the evader does not have

an incentive to deviate from its strategy to move directly to the predicted intercept

point.

Network abstraction

The above requirement by the application imposes a distance sensitivity network

abstraction that guarantees that complete information about nearby objects will be

available at a lower latency than farther objects.

We implement the distance sensitivity abstraction using network protocol Trail.

We use geometric ideas to design an energy-efficient, fault-tolerant and hierarchy-free

WSN service, Trail, that supports tracking-based WSN applications. The specifica-

tion of Trail is to return the location of a particular object in response to an in-network

8

subscriber issuing a find query regarding that object. Trail has a find cost that is lin-

ear (O(df)) in terms of the distance (df) of the subscriber from the object. To this

end, Trail maintains a tracking data structure by propagating mobile object infor-

mation only locally, and satisfying the distance sensitivity requirement for the track

updates. The amortized cost of updating a track when an object moves a distance

dm is O(dm ∗ log(dm)).

A basic Trail protocol can be refined by tuning certain parameters, thus resulting

in a family of Trail protocols. Appropriate refinements of the basic Trail protocol

are well suited for different network sizes and find/update frequency settings: One

refinement is to tighten its tracks by progressively increasing the rate at which the

tracking structure is updated; while this results in updating a large part of the tracking

structure per unit move, which is for large networks still update distance sensitive, it

significantly lowers the find costs for objects at larger distances. Another refinement

increases the number of points along a track, i.e., progressively loosens the tracking

structure in order to decrease the find costs and be more find-centric when object

updates are less frequent or objects are static. Moreover, Trail scales well to networks

of all sizes. As network size decreases, Trail gradually eschews local explorations and

updates and thus increasingly centralizes update and find.

We evaluate the performance of Trail by simulations in a 90×90 sensor network and

experiments on 105 Mica2 nodes in the Kansei testbed[3]. This implementation has

been used to support a distributed intruder interceptor tracking application where the

goal of the interceptor is to catch the intruders as far away from an asset as possible.

9

1.2.3 Classification of intruders and monitoring their tracks

The goal of this application is to classify and track objects such as humans, soldiers

(humans with metallic objects), small cars, SUVs and army vehicles at a centralized

base station. The network is equipped with sensors such as magnetometers, micro-

phones and motion sensors.

Application Logic

Techniques such as transmitting every sample sensed by individual nodes to the

base station or complex signal processing at nodes to perform classification may not

be feasible to implement in resource constrained sensor networks. Therefore we use a

technique by which each node only has to send a binary information upon ’detection’

in order to perform classification and tracking.

The influence field of an object j with respect to a given sensing modality is

the region surrounding j where it can be “detected” by a sensor of that type [10].

This region thus depends on both the characteristics of the object, such as its size and

shape, as well as the sensing modality being used. Differences in the area and/or shape

of the influence fields of different objects can be used to distinguish between them.

The influence field feature is thus useful in sensor network applications for surveillance,

where typical tasks include detection, classification, and tracking of various types of

objects. To estimate the influence field, each node merely has to detect a binary

presence of an object; network-based aggregation of these bits yields the influence field

without requiring substantial or complex node operation. The influence field feature

is thus well suited for wireless sensor network applications where individual nodes are

constrained due to limited processing, sensing and communication capabilities.

10

Estimation consists of calculating the area and the shape of the influence field.

If we assume sensor node density ρ exceeds some lower bound that depends on the

targets at hand, the estimation of the area A of the influence field of j is effectively

reduced to counting the number of nodes that detect j. The key challenge in realizing

the influence field is the unreliability of wireless sensor networks. We deal with both

node and network faults when estimating the influence fields. The requirement to

separate influence fields of different objects and track them, gives rise to the following

network abstractions.

We consider node faults of two types: false positives and false negatives can be

generated by a node regarding a particular detection due to many factors such as

environmental variations or node failures. We also consider two types of network

faults: channel fading and channel contention. We model the channel fading effect

as a constant probability of message loss over each hop. We model the channel

contention probability as a function of the number of nodes trying to send a message

simultaneously and the number of slots available.

Network abstractions and implementation

1. False report insensitivity: The net effect of a node fault is modeled as a node

missing a detection of an object, i.e. a false negative or a node asserting a de-

tection when there is no object, i.e. a false positive. Based on our experimental

observations for node faults across a wireless sensor network deployment, we as-

sume that failures at nodes occur independent of each other and of the objects.

This leads to the following abstraction False report insensitivity.

11

False report insensitivity implies that the network guarantees separation be-

tween estimated influence fields of objects despite false negatives and false pos-

itives.

Implementation: Insensitivity to false reports are implemented by identifying

a minimum density of deployment that ensures separation between influence

fields with high probability enough to satisfy application accuracy. In scenarios

where density cannot be met, we use temporal aggregation technique to ensure

separation between influence fields.

2. Distance insensitivity: Recall that the channel fading effect is modeled as a

constant probability of message loss over each hop. Therefore messages from

nodes closer to the aggregator have lower probability of failure. Hence the

estimated influence field of a small object close to the aggregator can overlap

with that of a large object far away. We therefore need to compensate for the

effect of distance of an object from the aggregator during its estimation.

Distance insensitivity implies that the network guarantees that the influence

field of an object estimated at the classifier is invariant with distance of the

object from the classifier.

Implementation: In order to achieve distance insensitivity, we use a probabilistic

reporting technique by which nodes report a detection with probability directly

proportional to their distance from the central classifier.

3. Contention insensitivity: As the number of nodes that report a detection in-

creases, there is increased channel contention leading to message losses. As a

result, after a point fewer detections are reported for larger objects compared

12

with smaller objects. We therefore need to compensate for the effect of influence

field size on the correct estimation.

Contention insensitivity implies that the network guarantees that the separation

of influence fields of objects is independent of the size of the influence fields of

the objects.

Implementation: In order to achieve contention insensitivity, we use the follow-

ing techniques. (1) Influence field suppression: In this technique, the influence

fields of all objects are suppressed by each node reporting only a fraction of

the actual reporting nodes. By doing this, the maximum influence field may

be decreased such that contention does not result in inversion. (2) Temporal

segregation: The influence field suppression may fail to avoid inversion if the

difference between the influence field of the smallest and largest object under a

given modality is so large that the influence field of the smallest object may not

be decreased any more. In such cases, we use temporal segregation by which

the reporting of detections are separated in time to avoid contention.

We then show how the different network abstractions can be composed to guaran-

tee separation of influence fields under a combination of faults. We also show how the

network abstractions can be used to achieve classification in the presence of multiple

objects. We experimentally validate the reliable estimation of influence fields by us-

ing the implementation of these abstractions to accurately classify and track persons,

soldiers, and vehicles in A Line in The Sand [2].

13

1.2.4 Distributed vibration control

The goal of this application is to sense and control vibrations in structures such as

a beam or a fairing structure used for launching of satellite payloads. The requirement

is to provide mission critical stability despite unreliabilities in the underlying network.

Application Logic

The system is modeled as a marginally stable linear time-invariant multi-variable

system S with m sensor-actuator pairs as shown below.

x = Ax + Bu (1.1)

y = Cx (1.2)

where x is an n-dimensional state vector [x1, x2, · · · , xn]T , u is an m-dimensional

actuator vector, B is an n×m dimensional matrix and the individual sensor-actuator

pairs are collocated. Since S is marginally stable, A has eigenvalues on the imaginary

axis. Since the individual pairs of sensors and actuators are collocated, we have the

following condition.

B = CT (1.3)

Starting at any state, without any control being applied the system maintains its

energy as it is marginally stable.

We apply the following local on-off output feedback control law to stabilize the

system.

ui = α × sign(yi), i = 1....m (1.4)

14

where α is less than zero. Further ui equals zero when yi is zero. Thus a correct

actuator can have 3 possible control values 0, −α and α. We choose |α|, the magnitude

of the actuator force, to be the maximum force that an actuator can apply and assume

that this is the same across all actuators.

Sensor actuator components being unreliable, the various component failures can

manifest themselves in the form of arbitrary actuator behavior in which case their

effect on the underlying systems can be severe. One of the methodologies for the

design of fault-tolerant control systems involves real-time fault detection, isolation

and control system reconfiguration [11, 39, 61, 44, 13]. An appropriate action is taken

after the diagnosis of the faults. This method still leaves the following challenges. The

hardware itself can be faulty causing the actuators to fail-stop and offer no control

or debond from their surface causing them to offer incorrect control. It is sometimes

also not feasible to integrate the fault detection, diagnosis and reconfiguration in

dynamical systems particularly when the available reaction time is limited. The

underlying fault detection service is itself vulnerable to faults in the middleware and

software services. This leads us to consider a Byzantine model for the actuator faults.

A Byzantine actuator can produce an arbitrary control input to the plant at all times.

Byzantine insensitivity network abstraction

The question then arises as to how can the control system be guaranteed to be

stable when a fraction of the actuators in the network are Byzantine. Byzantine

insensitivity implies that stability of the system is guaranteed despite a fraction of

the actuators being Byzantine.

15

Implementation of Byzantine insensitivity

Given a maximum of k Byzantine actuators, we first determine a necessary number

of actuators to guarantee asymptotic stability. Then for a 2 dimensional system, we

determine a sufficient number of redundant actuators and determine conditions on

placement of the actuators that will guarantee asymptotic stability of the control

system. We demonstrate our approach using a beam vibration control application as

a case study.

1.3 Organization of this Thesis

The rest of this thesis is organized as follows.

In Chapter 2, we describe a pursuer evader tracking application with requirement

for eventual catch, and describe a distributed snapshot service for wireless sensor

networks, that supports this application. In Chapter 3, we tighten the requirement

of the pursuer evader tracking application to satisfy an optimal catch and describe a

distributed sensor network based tracking service, Trail, that supports this applica-

tion. In Chapter 4, we design a surveillance application that classifies and monitors

the tracks of objects in a region. In Chapter 5, we describe the design of a reliable

vibration control application that guarantees asymptotic stability despite Byzantine

actuators. We discuss conclusions and future work in Chapter 6.

16

CHAPTER 2

DISTRIBUTED PURSUER EVADER TRACKINGAPPLICATION WITH EVENTUAL CATCH

Wireless sensor networks have enabled new surveillance systems, where sensor

nodes equipped with processing and communication capabilities can collaboratively

detect, classify and track targets of interest over a large area. These surveillance sys-

tems make it viable to use the state information collected through the sensor network

to guide mobile agents to achieve surveillance goals such as target capture and asset

protection. A sensor network surveillance system has the advantage of giving the

mobile agents access to the global information so that they can optimize their motion

for pursuit tasks, as opposed to resource-intensive search and map building tasks.

That said, using sensor networks to implement “active” surveillance strategies intro-

duces new challenges as well. Target track information obtained by local processing

of sensor information needs to be routed to mobile agents through multi-hop commu-

nication links, which results in delays, message losses and random arrival times of the

packets carrying track information. In addition, as the network is scaled in size, high

throughput rates for all pursuers cannot be sustained at all times, which necessitates

a network communication strategy that adapts to pursuer information requirements.

17

In this chapter, we consider a pursuer evader tracking scenario where multiple

pursuers and evaders exist in a bounded region. We design a pursuer control strategy

and associated network support that guarantees an eventual catch despite network

effects. We assume target information has been established through local fusion of

sensor data. This target information is communicated through the multi-hop wireless

network infrastructure to a pursuer object, which calculates the next move based on

evader and its own state. In summary, we show that eventual catch can be achieved

if network delays, update periods and error in the estimation of the target location

scale linearly with distance between the pursuer and the evader and describe a wireless

sensor network service that satisfies these conditions.

In Section 2.1, we describe the system model. In Section 2.2, we derive the suf-

ficient conditions for successful pursuit (that result in eventual catch) and translate

these to network abstractions. In Section 2.3, we describe a snapshot service for wire-

less sensor networks that implements these network abstractions. In Section 2.4, we

show how the snapshot service can be used to satisfy successful pursuit requirements.

In Section 2.5 we discuss work related to our snapshot service and pursuer evader

tracking applications. In Section 2.6, we present a summary.

2.1 System Model

In our system model one or more pursuer objects are interested in eventually

catching all evader objects spread over a bounded region. The pursuer objects are

assisted by a wireless sensor network that provides the state (location) of evader

objects to the pursuer objects. We assume that every pursuer is assigned to track

at most one evader at a time. We assume that all the objects in the network are

18

uniquely identifiable. A catch is said to occur when when the distance dpe between a

pursuer p and an evader e assigned to p is smaller than ε where ε approaches 0.

In our analysis we consider the case where one pursuer p has been assigned to an

evader e and this assignment holds until the evader is caught. Note that an eventual

catch in this scenario is sufficient to show that all evaders will be eventually caught.

Let vp and ve denote the pursuer and evader speeds respectively. The strategy of

pursuer p is as follows: given a location xe for the evader, follow the straight line to

xe.

2.2 Sufficient Conditions for Eventual Catch

If the network can provide the exact location of the evader e to pursuer p at all

times with no delay in transmission, then it is trivial that vp > ve is sufficient to

obtain eventual catch. In this section, we characterize tighter sufficient conditions for

an eventual catch.

Let z(t) denote the error in the location of evader at any time t. Let dpe(t) denote

the distance between the pursuer and evader at time t. Let δt denote the staleness

in the state of e supplied to p at time t. Let I(t) denote the maximum interval after

time t at which location of e can be provided to p. Let α = vp

veand α > 1.

Theorem 2.2.1. Evader e will be eventually caught by pursuer p if there exists con-

stant k > α+1α−1

and time To such that the following conditions hold at all t > To:

1. z(t) < dpe(t)

k

2. δ(t) < (dpe(t)

ve) ∗ (1 − α+k+1

α∗k )

3. I(t) < (dpe(t)vp

) ∗ (k+1k

)

19

Proof. Let lp denote the location of pursuer p at time t such that t > To. Let le be

the location of evader supplied to p at time t. If the maximum error and staleness in

this location satisfy conditions 1 and 2 stated above, the actual location le of evader

e at time t − δ(t) is within a ball of radius dpe(t)

karound le. At time t, evader e is

within a ball of radius dpe(t)k

+ δ(t) ∗ ve around le.

At time t, the action of the pursuer is to move towards le. Let us assume that next

information about the evader is available to p only after reaching le (or the pursuer

may even choose not to consume any information in this interval).

The maximum distance between lp and le is bounded by the following inequality.

dist(lp, le) < dpe(t) ∗ (k + 1

k) (2.1)

It follows using Eq. 2.1 that the maximum time I(t) to reach le is bounded by the

following inequality.

I(t) < (dpe(t)

vp

) ∗ (k + 1

k) (2.2)

Let dpe(t + t′) denote the distance between p and e at time t + t′. Using Eq. 2.2,

we have the following inequality.

dpe(t + I(t)) < (dpe(t)

vp

) ∗ (k + 1

k) ∗ ve +

dpe(t)

k+ δ(t) ∗ ve (2.3)

In order for eventual catch, we require that dpe(t + t′) < dpe(t). Using this, we get

the following inequality.

δ(t) < (dpe(t)

ve

) ∗ (1 − α + k + 1

α ∗ k) (2.4)

20

Note that for δ(t) > 0, we require that α + k + 1 > α ∗ k. Using this we get the

following condition:

k >α + 1

α − 1(2.5)

Note that α > 1.

Using the above theorem, we showed that error, latency and update periods that

decrease linearly with distance are sufficient to achieve eventual catch in pursuer

tracking application. We now define 3 network abstractions that capture these re-

quirements and describe a network service that implements these abstractions.

2.3 Snapshot service for wireless sensor networks

In this section, we systematically design wireless sensor network algorithms that

periodically deliver distance sensitive snapshots to all nodes in the network. Our

algorithms are easily adapted to allow snapshots to be delivered only to a subset

of nodes as opposed to all nodes. They are memory efficient, requiring only O(3f ∗

log(N1f ) ∗ m) bits per node. They are readily realized in networks with irregular

density, networks with arbitrary sized holes, imperfect clustering, and non unit disk

radios. We quantify the maximum rate at which information can be generated at each

node so that snapshots are periodically delivered across the network, the algorithms

can of course be operated at lower rates than these. For our services, global time

synchronization is not required; a local notion of time however is needed to ensure

fair scheduling of transmission of nodes.

21

Overview of algorithms and main results

Consider an ideal network where nodes are embedded in a virtual 2-d grid such

that there is exactly one node at each grid location and that each grid node can

reliably reach each of its neighbors in the grid and no others. We first describe

an algorithm to deliver snapshots with distance sensitive resolution and latency at

all nodes at the maximum rate. To obtain distance sensitive resolution, instead of

dispersing the individual state of each node, we map the state of nodes into aggregate

values of non-overlapping regions. We then deliver snapshots across the network such

in a snapshot delivered to a node j, the size of a region into which a node i is mapped

increases as dist(i, j) increases. Thus, the resolution with which i is represented in

the snapshot decreases as dis(i, j) increases. To achieve this kind of snapshot delivery,

we refine the clustering of nodes into a hierarchical one with a logarithmic number of

levels as the network size. The basic idea is that a clusterhead at each level compresses

data from all nodes in that level into m bits. Thus, the data aggregated at each level

is represented by the same number of bits. At higher levels, the data is summarized

with a coarser resolution as these levels contain more nodes.

In order to ensure uniform distance sensitive latency, we regulate the flow of in-

formation in all directions by proceeding in rounds. Intuitively, a round is a unit of

time when information is exchanged between any level 1 cluster and all its neighbor-

ing clusters. Our scheduling and other protocol actions at each step are such that

information is propagated across the network in a pipelined manner; by this, new

information can be generated at a node as soon as previous information has been

dispersed only to its local neighborhood as opposed to the entire network.

22

In the first algorithm, in a snapshot S of the network delivered to node jthe resolution of the state of a node i in S decreases as O(df), the staleness

of the state of a node i in S is O(32f ∗ m ∗ log(n) ∗ d) and the averagenetwork communication cost for N samples is O(3f ∗ log(n) ∗ N ∗ m).

To achieve distance sensitive rate, we schedule the delivery of aggregated infor-

mation at each level such that information of higher levels is delivered over a larger

interval as opposed to lower levels. To do this, we allocate an exponentially increasing

number of bits per message to lower level aggregates so that they are delivered at a

faster rate.

In this second algorithm, the average communication cost per N samples(one from each node) is O(3f∗N∗(m+log(n/m))), the maximum staleness

in the state of a node i received by a snapshot at node j is O((m +

log(n/m)) ∗ d) where d = dist(i, j) and the maximum interval betweenwhen a node j receives the state of node i is O((m + log(n/m) ∗ d).

Our algorithms allow for a user-pluggable aggregation function. We require only

that the function, say f, be idempotent and satisfy the following decomposability

property: ∀a, b, f(a ∪ b) = f(f(a) ∪ f(b)). Examples of such functions are average,

max, min, count and wavelet functions.

We then relax our regularity assumptions and describe how our algorithms handle

the cases of non uniform density, non uniform radio range and holes of arbitrary sizes

in the network. The case of over density is modeled as certain virtual grid locations

containing more than 1 node. In the case of holes in the network, we show that our

algorithms achieve distance sensitivity in terms of the shortest communication path

between any two nodes as opposed to the physical distance.

23

2.3.1 Network model and problem statement

In this section, we present the system model and a generalization of the concept

of snapshots based on distance sensitive properties.

Network model

The sensor network consists of N nodes that are embedded in an f -dimensional

space. We let n abbreviate N1f . The nodes induce a connected network where each

communicate at W bits per second. Nodes are synchronized in time. Each node j

periodically generates m bits of (sensor) information, and maintains a data structure

comprising the most recent state of nodes (or partitions of nodes) and a timestamp

associated with that state. We let j.Xi and j.Ti (respectively, j.Xp and j.Tp) refer to

the state of node i (respectively, partition p) and the timestamp of that state at node

j.

For ease of exposition, we restrict our attention to sensor networks that form

a 2 dimensional grid with a node at every grid location. We further assume that

node communication follows an idealized disk model: specifically, each node can

communicate reliably with all its neighbors in the grid and with no others. We define

the neighbors of node j to be the ones to its north, east, west, and south and also

to its northeast, northwest, southeast, or southwest that exist in the grid; we denote

these (up to 8) neighbors as j.n, j.e, j.w, j, s, j.ne, j.nw, j.se and j.sw respectively.

In Subsection 2.3.4, we remove each of these restrictive assumptions.

Generalized snapshots

Let’s begin by considering global snapshots where state is maintained for each

node.

24

Definition 1 (Snapshot S). A snapshot S is a mapping from each node in the network

to a state value and a timestamp associated with that state value.

A consistent snapshot is one where the timestamps associated with each state

value are all the same. The staleness of a state value in S is the time elapsed between

its timestamp and the current time. We now consider a generalization where state

values do not necessarily correspond to the same instant of time but their staleness

enjoys a distance sensitive property.

Definition 2 (Snapshots with distance sensitive latency). A snapshot S received by

a node j has distance sensitive latency if the staleness in the state of each node i in

S decreases as dist(i, j) decreases.

We now further generalize the notion of snapshots so that state is associated with

partitions p of the network as opposed to individual nodes. Let P be a partitioning

of the network.

Definition 3 (Snapshot S of P ). A snapshot S of P is a mapping from each partition

p in P to a state value and a timestamp associated with that state value.

The generalized definition is useful even if P is not a total but a partial partition,

i.e., not all nodes are represented in the snapshot. More to the point, the state and

timestamp of each p in S intuitively represent the aggregate state of all nodes in the

partition and the aggregate timestamp. We assume that the timestamp of recoding

the state of all nodes in any partition p is the same, and refer to this common value as

the aggregate timestamp. Note that the aggregate timestamp of different partitions

may be different.

25

As there may not exist a mapping from the aggregate state of a partition to

the exact state of individual nodes that was recorded for the purpose of computing

the aggregate, the latter may be estimated using some function of the state of the

partition. The resolution of the state of a node in a snapshot is an inverse measure

of the error between the state of the node that was recorded and the aggregate state

of the partition p that it belongs to.

We are interested in snapshots where the increase in the error in the state of

a node is bounded by the size of the partition p to which it belongs and thus the

decrease in resolution of the state of a node is bounded by the size of p. This leads

us to consider a generalization where the resolution of the state of a node increases

as distance decreases.

Definition 4 (Snapshots with distance sensitive resolution). A snapshot S of P

received by a node j has distance sensitive resolution if the resolution of the state of

each node i covered by P increases as dist(i, j) decreases.

Informally speaking, the size of the partition to which the state of node i is

mapped into in a snapshot received at j increases as dist(i, j) increases. Therefore

the resolution with which i is represented in S decreases with distance. In other

words, j has a myopic or fisheye view of the network.

Finally, we consider a generalization where the rate at which state of the nodes is

reported to a node decreases as the distance of the nodes increase.

Definition 5 (Snapshots with distance sensitive rate). A node j receives snapshots

of P with distance sensitive rate if the rate at which the state of each node i covered

by P is updated in snapshots received by j increases as dist(i, j) decreases.

26

Note that the concept of distance sensitive rate is orthogonal to that distance

sensitive latency. In the latter staleness in the state received decreases with distance

but fresh information arrives at the same rate at all nodes, where as in the former,

the state of nearby nodes is reported more often than farther nodes.

2.3.2 Distance sensitive resolution and latency

We partition the network into a hierarchical one with a logarithmic number of

levels, which are numbered 0..(log3n). A 3 by 3 set of 9 level r clusters form a cluster

at level r + 1, as illustrated in Fig. 2.1. Each node belongs to one cluster at each

level, and each cluster has a clusterhead which is the center node of that cluster. A

clusterhead at level r is also a clusterhead at levels 0..r − 1.

Overview of algorithm

The basic idea is that a clusterhead at each level compresses data from all nodes

in that level into m bits. Thus, aggregated data at each level is represented by the

same number of bits. At higher levels, data is summarized into a coarser resolution

as the levels contain more nodes. The aggregated data is then dispersed to all nodes

at that level. This solution suffers from a multi-level boundary problem however: two

nodes could be neighbors but belong to a common cluster only at level r 1. Thus

despite being neighbors, both nodes get a summary of the other at a much coarser

resolution than desired. The multi-level boundary problem is illustrated in Fig. 2.1,

where nodes j and k are neighbors at level 0 but belong to a common cluster only

at level 3. To avoid this problem, we disperse a summary computed at level r not

only to nodes in level r cluster, but also to nodes in all neighboring level r clusters.

This algorithm can be implemented in multiple phases; aggregates at each level are

27

computed and dispersed sequentially. When implemented sequentially, however, new

samples can be generated at each node only after data is dispersed at all levels. To

avoid this constraint, we use the following pipelined implementation described next.

Notations

Let j.L be the highest level for which j is clusterhead. Note that there are at

most 8 neighbors at each level for each node in the grid topology. We implement

virtual trees along the structure at each level. To describe these trees, we will need

the following definitions.

Figure 2.1: Hierarchical clustering

28

Figure 2.2: Neighboring level 1 clusters

Definition 6 (tree(k, j)). tree(k, j), where j is a level k clusterhead, is a level k tree

formed with j as root and spanning all nodes in the level k cluster of j and all level

k clusters that are its neighbors.

Definition 7 (j.in(k, y)). For each tree(k, y) that j belongs to, j.in(k, y) is j’s parent

towards root y.

Definition 8 (j.out(k, y)). For each tree(k, y) that j belongs to, j.out(k, y) is the set

of j’s descendants on the tree.

Definition 9 (M(k, y)). M(k, y) is the level k summary computed by a level k

clusterhead y.

In Fig. 2.3, a level 1 tree rooted at j is shown as an illustration. The level 1 tree

extends up to all level 0 nodes in its own cluster and level 0 nodes in the 8 neighboring

level 1 clusters. The trees represent the distance up to which an aggregate at any

level is propagated.

Schedule

We schedule the nodes to transmit in rounds. A round is a unit of time in which

information is exchanged between a level 1 clusterhead and all of its 8 neighboring

29

level 1 clusterheads. Each round is divided into multiple slots. In the first slot, all

level 1 clusterheads transmit. In the remaining slots, all level 0 nodes in each cluster

transmit twice. The second transmission by a node within a round takes place after

all its 8 neighbors have transmitted at least once. Intuitively speaking, during the

first turn for a node, information is communicated outwards from the clusterhead. In

the next turn for the node, information is communicated to the level 1 clusterhead

that the node belongs to. A simple non-interference schedule that satisfies these

constraints is one where all level 0 nodes take turns in say a clockwise direction. For

example, level 0 nodes transmit in a clockwise direction starting from j.ne, where j

is a level 1 node. Each round thus consists of 17 slots.

Local storage

Each node i stores the most recent value of M(x, y) received by i for each

tree(x, y) that i belongs to. The state of any node j is obtained as a function of

M(x′, y′) where x′ is the smallest level that contains information about j. Recall that

the resolution of the state of j decreases as the number of nodes in the aggregate

M(x′, y′) increases.

Figure 2.3: Illustrating level 1 tree rooted at j

30

Algorithm S1

We describe the actions executed by the nodes in three parts: (1) send / compute

actions for nodes with j.L > 0, (2) send / compute actions for nodes with j.L = 0

and (3) receive / update actions for all nodes.

• In slot 0 of each round nodes with j.L > 0 compute the summary M(r, j) for

each level 1 ≤ r ≤ j.L that they are a clusterhead of based on the corresponding

lower level information received in the previous round. The computed summary

at each level is transmitted to the children on the respective tree rooted at j.

Thus M(r, j) is sent to j.out(r, j) for 1 ≤ r ≤ j.L. In addition, for each

tree(x, y) that j.L belongs to but is not a root of, transmit M(x, y) as heard in

slot 0 from j.in(x, y) to j.out(x, y).

• To explain the actions of level 0 nodes, without loss of generality, consider level

1 nodes j and k and level 0 nodes j.ne and k.sw as shown in Fig. 2.2.

– In slot 1, for each tree(x, y) that j.ne belongs to but is not a leaf of,

transmit M(x, y) as heard in slot 0 from j.in(x, y) to j.out(x, y). Also,

transmit its own information M(0, j.ne) to children in the level 0 tree

rooted at j.ne.

– In slot 9, for each tree(x, y) that j.ne belongs to but not a leaf of, transmit

M(x, y) as heard in slots 2 to 8 from j.in(x, y) to j.out(x, y).

• The action at any node j upon receiving a message from i is as follows: for each

tree(x, y) that j belongs to, store M(x, y) if i = j.in(x, y).

31

In summary, aggregates computed at each level are copied only going downwards

along a tree. This is sufficient for a level r node to compute aggregates from level

r−1 nodes, because a tree at level r−1 extends up to all level 0 nodes in neighboring

level r − 1 clusters. And one of the neighboring level r − 1 node is a level r node.

Thus, when a computed aggregate by any node is being dispersed to nodes in its own

cluster and the neighboring clusters, it is also being sent in to a higher level node to

compute an aggregate. In Fig. 2.3, nodes p and q are level 2 clusterheads. Note that

the level 1 tree rooted at j reaches the level 2 clusterhead q that j belongs to. Since

a level r node is equidistant from all level r − 1 nodes, the computed summaries are

synchronous. We now analyze the latency and communication cost of algorithm S2.

Analysis

Lemma 2.3.1. In S1, the maximum message length needed per slot is (9 ∗ log(n) −

7) ∗ m bits.

Proof. Consider a node j at any level r, where 0 ≤ r ≤ log(n). Let 1 ≤ l ≤ log(n)−1.

A level l tree rooted in the same level l cluster as that of j can pass through j. A level

l tree rooted in neighboring level l clusters as that of j can also pass through j, but

no other level l tree can pass through j. Thus, at most 9 trees at levels 1..log(n) − 1

can pass through each node. There is only one level logn tree. Also j belongs to only

one level 0 tree for which it is not a leaf. The maximum message length needed per

slot in algorithm S3 is (9 ∗ log(n) − 7) ∗ m bits.

It follows that the slot width sw needed in algorithm S1 is (9∗log(n)−7)∗mW

bits per

second.

32

Lemma 2.3.2. In S1, the maximum staleness in the state of a node i received by a

snapshot at node j is O(log(n) ∗ m ∗ d) where d = dist(i, j).

Proof. Consider a node p at level r. To compute a summary at level r, level r − 1

summaries are needed. dist(p, q) = 3r−1, where q is any node in the set p.nbr(r).

Thus, a level r summary is computed based on a level r − 1 summary that was

generated (17/3) ∗ 3r−1 ∗ sw time ago, since latency between each pair of level 1 nodes

is 17 slots. A level r−1 summary is computed based a level r−2 summary, and so on

until level 0. Upon summation, we see that the staleness of a level 0 (individual node)

state information in a level r summary is equal to (17/2) ∗ 3r−1 ∗ sw. The maximum

distance traveled by a level r summary is (3/2)∗3r. The maximum latency involved is

(17/2) ∗ 3r ∗ sw. The minimum distance between j and i for which a level r summary

is the smallest level that contains information about j is 3r−1.

The maximum staleness in the state of a node i at node j is given by the following

equation:

staleness(i, j) = (L1 + L2) (2.6)

=(L1 + L2)

3r−1× 3r−1 (2.7)

= O(34 ∗ sw ∗ d) (2.8)

= O(log(n) ∗ m ∗ d) (2.9)

The result follows.

Lemma 2.3.3. In S1, the resolution of state of a node i in a snapshot received at

node j is Ω( 1d2 ) where d = dist(i, j).

33

Proof. In a level r summary, the state of 9r nodes is compressed into m bits. We

thus regard the error in the state of each node in that summary to be O(9r). The

minimum distance between i and j at which j gets a level r summary of i but not

a level r − 1 summary of i is 3r−1. Thus, the error in the state of i in a snapshot

received at j is O(d2) and the resolution of state of i in a snapshot received at j is

Ω( 1d2 ), where d = dist(i, j).

Lemma 2.3.4. In S1, the average communication cost in the network to deliver a

snapshot of one sample from each node to all nodes is O(N ∗ log(n) ∗ m).

Proof. To deliver a snapshot with a sample from each node, every node communicates

O(m∗ log(n)) bits n times. And to deliver a snapshot with y samples from each node,

every node communicates O((n + y) ∗ (m ∗ log(n))) bits, since all the y samples are

pipelined. Hence, if y is large and y = Ω(n), the average communication cost at each

node to deliver a snapshot of a sample from each node to all nodes is O(m ∗ log(n)).

The average communication cost over N nodes is O(N ∗ (m ∗ log(n)).

Note that if y is small, for instance, if there is only one sample from each node,

then the communication cost is O(N ∗n ∗ (m + log(n/m)). Pipelining the delivery of

snapshots improves the average communication cost to O(N ∗ (m ∗ log(n)).

Lemma 2.3.5. In S1, the memory requirement per node is O(log(n) ∗ m) bits.

Proof. Recall that the data structure maintained at each node is the most recent

value of M(x, y) received by i for each tree(x, y) that i belongs to. Nodes do not

buffer information to be forwarded over multiple rounds. The maximum number of

trees through any node is O(log(n)), with m bits of information flowing along each

tree. The result follows.

34

Extending to other dimensions

Consider an f dimensional structure. In this structure, nodes are divided into

clusters with 3f nodes per cluster. Thus there are 3f − 1 level 0 nodes per cluster.

Each round consists of 2 ∗ 3f − 1 slots. The number of slots per round increases

proportional to 3f .

Using these, we generalize Lemma 2.3.2 and Lemma 2.3.4 as follows:

Lemma 2.3.6. In S1, the maximum staleness in the state of a node i received by

a snapshot at node j in a network of f dimensions is O(32f ∗ log(n) ∗ m ∗ d) where

d = dist(i, j).

Lemma 2.3.7. In S1, the average communication cost in the network to deliver a

snapshot of one sample from each node to all nodes is O(N ∗ 3f ∗ log(n) ∗ m).

Note also that in an f dimensional structure, the state of 3f∗r nodes is compressed

in m bits. Following the structure of proof for Lemma 2.3.3, we get the following

result.

Lemma 2.3.8. In S1, the resolution of state of a node i in a snapshot received at

node j in an f dimensional network is Ω( 1df ) where d = dist(i, j).

2.3.3 Distance sensitive rate

In this subsection, we describe two algorithms in which nodes receive snapshots

that are distance sensitive in latency, resolution and also distance sensitive in rate.

We partition the network hierarchically into clusters and schedule nodes to trans-

mit in rounds exactly as we did in algorithm S2. The snapshot at each level is

aggregated into m bits. However, instead of transmitting m bits for each level of data

35

in every round, we allocate the number of bits hierarchically. Accordingly, a message

transmitted by a node in any given round consists of m bits for each level 0 informa-

tion, m/3 bits for each level 1 information, and 1 bit for each level from log(m) to

log(n). The actions executed by every node in their corresponding slots remain the

same as in algorithm S2 except that every level r summary is now transmitted over

min(3r, m) rounds, as opposed to in each round containing a new level r summary.

Algorithm S3a

By way of refining algorithm S2, consider a level 0 node with j.L = r. A level

r summary is computed by this node once every 3r rounds based on the most recent

level r − 1 summaries it receives. This summary M(r, j), which consists of m bits,

is transmitted in slot 0 of each round with max(1, m3r ) bits per round. Thus, a level

r summary is sent over min(3r, m) rounds. The actions for forwarding nodes remain

the same except for the change that each node now only receives a fraction of M(x, y)

in every round for each tree(x, y) that it belongs to, and it forwards only that fraction

in the next round. We now analyze the latency and communication cost of algorithm

S3a.

Analysis

Lemma 2.3.9. In S3a, the maximum message length needed per slot in algorithm is

11 ∗ m2

+ 9 ∗ log( nm

) bits.

Proof. Recall that at most 9 trees at levels 1..log(n)− 1 can pass through each node,

and there is only one level logn tree. Also, j belongs to only one level 0 tree for which

it is not a leaf. Moreover, m3r bits are allocated for each level 0 ≤ r ≤ log(m) and 1

36

bit for each level z where log(m) < z ≤ log(n). The maximum message length (ML)

needed is obtained by the following equation:

ML =m

30+

i=log(m)∑

i=1

(9 ∗ m

3i) + 9 ∗ (log(n/m)) (2.10)

= m + m ∗ 9

2∗ (1 − 1

m) + 9 ∗ log(

n

m) (2.11)

< m + m ∗ 9

2+ 9 ∗ log(

n

m) (2.12)

The result follows from these facts.

It follows that the slot width sw needed in algorithm S3a ism∗ 11

2+9∗log( n

m)

Wbits per

second.

Lemma 2.3.10. In S3a, the maximum staleness in the state of a node i received by

a snapshot at node j is O((m + log(n/m)) ∗ d) where d = dist(i, j).

Proof. Consider a node p at level r where r ≤ log(m). To compute a summary at

level r, level r − 1 summaries are needed. dist(p, q) = 3r−1 where q is any node in

the set p.nbr(r). The latency between each pair of level 1 nodes is 17 slots. Thus the

latency to travel from level r − 1 node to level r node is given by (17/3) ∗ 3r−1 ∗ sw.

But in this algorithm, a level r − 1 summary is actually transmitted in 3(r − 1)

rounds by dividing it into 3r−1 parts. Thus, a level r summary is computed based

on level r − 1 summary that was generated 2 ∗ (17/3) ∗ 3r−1 ∗ sw time ago. A level

r − 1 summary is computed based a level r − 2 summary and so on until level 0.

Upon summation, we see that the staleness of a level 0 state information in a level r

summary is (17) ∗ 3r−1 ∗ sw.

Note that a complete level r − 1 snapshot is sent every 3r−1 rounds in a pipelined

manner. Thus every 3r−1 rounds, a level r − 1 snapshot is received by a node. On

37

the other hand a level r snapshot is computed only every 3r rounds. Thus a fresher

level r − 1 snapshot is always available to compute a new level r snapshot.

The maximum distance traveled by a level r summary is (3/2) ∗ 3r. However, this

summary is sent in 3r rounds. The maximum latency involved is 2 ∗ (17/2) ∗ 3r ∗ sw.

Recall that in order to update the local data structure of j, the state of a node

i is updated using summary M(x′, y′) where x′ is the lowest level which contains

information about k. Now the minimum distance between j and i for which a level r

summary is the smallest level that contains information about j is 3r−1.

The maximum staleness in the state of a node i at node j is given by the following

equation:

staleness(i, j) = (L1 + L2) (2.13)

=(L1 + L2)

3r−1× 3r−1 (2.14)

= O(68 ∗ sw ∗ d) (2.15)

= O(m ∗ d + log(n/m) ∗ d) (2.16)

Note that at levels r > log(m), 1 bit is allocated per round. Thus, for all these

levels, a summary can be sent out in less than 3r rounds. The maximum staleness

for those nodes whose state is obtained using summaries greater than level r is less

than that derived in the above equation.

The maximum staleness in the state of a node i received by a snapshot at node j

is thus O((m + log(n/m)) ∗ d), where d = dist(i, j).

Lemma 2.3.11. In S3a, the maximum interval between when a node j receives the

state of node i is O((m + log(n/m) ∗ d), where d = dist(i, j).

38

Proof. Consider levels 0 ≤ r ≤ log(m). Note that a complete level r snapshot is sent

every 3r rounds in a pipelined manner. Thus every 3r rounds, a level r snapshot is

received by a node. The time corresponding to 3r rounds is (17/3) ∗ 3r ∗ sw.

Recall that in order to update the local data structure of j, the state of a node

i is updated using summary M(x′, y′) where x′ is the lowest level which contains

information about k. Now the minimum distance between j and i for which a level r

summary is the smallest level that contains information about j is 3r−1.

The maximum interval between when a node j receives the state of node i is given

by the following equation:

interval(i, j) =(17/3) ∗ 3r ∗ sw

3r−1× 3r−1 (2.17)

= O(17 ∗ sw ∗ d) (2.18)

= O(m ∗ d + log(n/m) ∗ d) (2.19)

Note that for levels r > log(m), 1 bit is allocated per round. Thus, for all these

levels, a summary can be sent out in less than 3r rounds. The maximum interval

between receiving two successive state information for those nodes whose state is

obtained using summaries greater than level r is less than that derived in the above

equation. The maximum interval between when a j receives the state of i is thus

O((m + log(n/m) ∗ d).

Lemma 2.3.12. In S3a, the average communication cost to deliver a snapshot of

one sample from each node to all nodes is O(N ∗ (m + log(n/m)).

Proof. To deliver a snapshot corresponding to 1 sample from each node, every node

communicates O(m+log(n/m)) bits n times. And to deliver a snapshot corresponding

39

to y samples from each node, every node communicates O((n + y) ∗ (m + log(n/m)))

bits since all the y samples are pipelined. Hence if y is large and y = ω(n), the

average communication cost at each node to deliver a snapshot of one sample from

each node to all nodes is O(m + log(n/m)). The average communication cost over N

nodes is O(N ∗ (m + log(n/m)).

Lemma 2.3.13. In S3a, the memory requirement per node is O(log(n) ∗ m).

Proof. The maximum number of trees passing through any node is O(log(n)). The

storage at each node is the most recent M(x, y) for each tree(x, y) that a node belongs

to. In algorithm S3a, the difference is that this information ((M(x, y)) arrives at a

node over multiple rounds. The memory requirement is still O(log(n) ∗m). We note

that although information received by a node in a given round is buffered for certain

number of rounds in schemes S3a, fresh information about a level is stored only after

existing information about a level is disseminated. The maximum information that

can be buffered by any node for a given level is still log(n) ∗ m. Hence we have the

result.

Algorithm S3a can be generalized to f dimensions just as algorithm S2 is. We

summarize our results in Fig. 2.4, which shows the bounds on staleness and resolution

in the state of i at nodes that are distance d away, the bound on interval at which

state updates of i are received, and the average communication cost for delivering a

snapshot of N samples (one from each node) across the network.

Discussion

We note that apart from decreasing the average communication cost per sample

from each node, solution S3a also offers flexibility in terms of the size of each sample.

40

Algorithm Staleness Communication cost Resolution Interval MemoryS2 O(32f ∗ log(n) ∗ m ∗ d O(3f ∗ N ∗ m ∗ log(n)) Ω( 1

df ) independent of d 3f ∗ log(n) ∗ mS3a O(32f ∗ (m + log(n/m)) ∗ d) O(3f ∗ N ∗ (m + log(n/m))) Ω( 1

df ) O(3f ∗ (m + log(n/m)) ∗ d) 3f ∗ log(n) ∗ m

Figure 2.4: Summary of results for snapshot algorithms

When m << log(n), the factor log(n) dominates and the average communication cost

is O(N ∗ log(n)) and the staleness is bounded by O(log(n) ∗ d). In algorithm S2 the

corresponding costs are O(N ∗ m ∗ log(n)) and O(m ∗ log(n) ∗ d) respectively. But

when the sample size increases to as large as order n, the average communication cost

is O(N ∗ n) and the staleness is bounded by O(n ∗ d), where as in algorithm S2 the

corresponding costs are O(N ∗ n ∗ log(n)) and O(n ∗ log(n) ∗ d) respectively.

2.3.4 Irregular Networks

In this subsection, we relax several assumptions we have made in designing al-

gorithms S1 − S2. We show how the algorithms continue to yield distance sensitive

snapshots and quantify the impact on the performance in the following cases: non

uniform density, holes of arbitrary sizes within the connected network, non unit disk

radios and imperfect clustering.

Clustering Model

We assume the existence of a clustering layer that partitions the general but con-

nected network, as modeled in Section 2, into hierarchical clusters such that every

network node belongs to one cluster at each level. As perfect (i.e., regular and sym-

metric) clustering may no longer be possible, we may not assume, for example, that

all level 0 nodes are within 1 hop range of the level 1 clusterhead. We weaken that

assumption to: each level 1 cluster includes all nodes that are 1 hop away but may

41

also include nodes that are up to some bounded number of hops, z, from it. Likewise,

all higher level clusterheads also have the same radius range as opposed to a uniform

radius. Moreover, the paths between neighboring clusters also need not the shortest

ones, unlike what we assumed in the previous sections.

More formally, our clustering assumption is stated as follows (with distance stand-

ing for communication hop distances):

• (C1) All nodes within distance 3k−12

from a level k clusterhead belong to that

cluster.

• (C2) The maximum distance of a node from its level k clusterhead is zk × 3k−12

.

• (C3) There exists a path from each clusterhead to all nodes in that cluster

containing only nodes belonging to that cluster.

• (C4) At all levels k > 0, there is at least one and at most 8 neighboring level

k clusters for each level k clusterhead and there exists a path between any two

neighboring clusterheads.

We note that the existence of such clustering solutions has been validated in

previous research [57] and also been used in the context of object tracking.

Networks with non uniform density

Once the network has been partitioned into clusters, we impose a virtual grid on

the network, as shown in Fig. 2.5. Each level 0 node belongs to some cell, but now

each cell in the virtual grid may contain any number of nodes. In particular, cells may

be empty and empty cells may be contiguous; we call sets of contiguous empty cells

the holes of the network. We first describe how the case of over density is handled.

42

Figure 2.5: Virtual grid and cells with different densities

Over density cells: In the virtual grid, each cell gets a slot to transmit as

described in algorithms S0− S1. When a cell has more than one node, each node in

the cell gets a turn over multiple rounds to send its data, resulting in time sharing

between nodes of a cell to transmit its own data. However, once data is sent out from

the source, the forwarding of the data does not incur this extra delay despite going

through denser cells. This is because any node in the dense cell that gets a turn in a

given round can forward the data heard in the previous round from neighboring cells.

Under density cells, holes, and imperfect clustering: If a particular cell

is empty in sparse regions of the network, then the communication path between

neighboring level k clusters (0 < k < log(n)) is lost.

Scheduling scheme (FS): Recall that a round is a unit of time in which infor-

mation is exchanged between a level 1 clusterhead and all its neighboring level 1

clusterheads. In the general model, a level 1 cluster can cover up to a z hop neigh-

borhood. Accordingly, the basic round scheduling introduced in Section 2 is adapted

as follows. For ease of explanation, assume z = 2. Thus, a level 1 cluster comprises

a minimum of the 1 hop neighborhood and a maximum of the 2 hop neighborhood.

At the beginning of each round, level 1 clusterheads transmit. There can be at most

43

26 level 0 nodes in each cluster. Some may be absent because of holes or because of

non uniform clustering. Depending on the position in the cluster, each level 0 node

gets 2 slots per round such that information is exchanged from all 8 directions. The

round length thus increases with z.

Distance sensitivity: A level k tree rooted at any node j spans all nodes in its

own level k cluster and all nodes in the neighboring level k clusters, using fixed paths

as described in Sections 3 and 4. If a neighboring level k cluster is completely absent,

then there is no need to reroute the information. However if the neighboring level k

cluster is present but the path is broken, then information has to be re-routed. We

now investigate the impact on latency and resolution when the information takes a

different path than normal.

Recalling the clustering specifications stated above, consider any two nodes i and

j in the network. Let the shortest path between these two nodes in the presence of

holes be p.

Lemma 2.3.14. If k is the smallest level at which i and j are neighbors then p > 3k−1.

Proof. Note that i and j are not neighbors at level k − 1. And if p ≤ 3k−1, then a

level k − 1 cluster cannot exist between i and j since from property C1, a level k − 1

cluster has a minimum radius of 3k−1−12

.

Theorem 2.3.15. Under CM , lgorithms S1, S2 yield snapshots that retain their

distance sensitive properties (Since network is not regular, the distance sensitivity is

strictly hop distance sensitivity and does not translate to geometric distance).

44

Proof. From the previous lemma, the minimum distance between two nodes i and j

for which level k is the smallest level at which i and j are neighbors is 3k−1. Using

C3 and C4 we have that level k information can be exchanged between i and j.

Despite the fact the trees are not formed along the regular grid pattern, it still

holds that not more than 9 trees per level pass through any node. This is because

there at most 8 neighboring level k clusters for any level k cluster. Moreover, the

maximum degree of any node in all trees is still 8, by imposing the virtual grid for

level 0. Therefore, the slot width allocations in algorithms S1, S2, S3a and S3b are

sufficient to transmit all information despite the trees not being created in a regular

pattern.

Figure 2.6: Handling holes in dense networks

Rerouting in Case 1 is illustrated in Fig. 2.6. The figure shows a level 1 cluster

with a clusterhead A that has 7 neighboring level 1 clusters. The small unfilled circles

represent cells of the virtual grid; these may contain one or more level 0 nodes. The

level 1 clusters cover up to a 2 hop neighborhood. The figure also shows a level 1 tree

rooted at A and extending up to clusters B and C.

45

Non uniform radio range

The design of algorithms S0−S3 assumed a strictly 1 hop communication range.

If communication range were relaxed to radio interference range varying from 1 to

s hops, the basic scheduling for each round would need to take into account this

additional interference. This would result in longer round lengths proportional to the

size of interference region.

Implementation considerations

In this subsection, we highlight considerations for implementing our snapshot

services in wireless sensor networks. In the past, we have implemented Trail [46], an

asynchronous network protocol for querying object states and used this in the context

of a distributed object tracking application. We have noted that as the objects in the

network get densely located, interference issues lead to high latency and decreased

efficiency. The snapshot services that we consider in this paper are high density

operations and TDMA is naturally suited for such scenarios as interference can be

avoided. But we do not need global time synchronization in the network. Nodes in

their network can learn their TDMA slots by knowing their relative position to that

of a clusterhead and locally scheduling in a non interference manner. Another issue

to consider is that of localization. For our snapshot services, knowledge of relative

location with respect to clusterheads is sufficient as opposed to the knowledge of

precise coordinates.

46

2.4 Using the snapshot service for distributed object track-ing

In the previous section, we described a snapshot service for wireless sensor net-

works that provided distance sensitive latency, resolution and rate. In this section,

we describe how this service can be used for distributed tracking. Specifically, we

describe how the location information for objects can be compressed such that we

have an error that grows only linearly with distance as is required by the tracking

application.

Let the number of bit m allocated for information at each level be equal to the

number of evaders in the network. Recall that the identifiers of the evaders are known.

Every clusterhead at level i sets the bit e in this information if object e is within its

cluster. Thus the highest resolution at which the locatio of an evader is available is

the size of the level 0 cluster. We relax our condition for catch to be such that the

pursuer and the evader are in the same level 0 cluster.

Note that if the smallest level at which information about an evader e is available

to pursuer p is k, then it must be the case that dpe is at least 3k. Thus, the maximum

error in the estimation of distance to e is bounded by the radius of the level k cluster,

i.e. 3k

2. Thus, we get that the maximum error is bounded by dpe

2when distance

between p and e is dpe. Thus we obtain an error that grows at most linearly with

distance between p and e and is always less than the distance between p and e and

satisfying conditions for optimal pursuit. We have already shown that latency grows

as O(log(n) ∗ d) even when information is updated in every round. Thus using the

snapshot service, information about e is available at a faster rate than required.

47

Thus we see that our distance sensitive snapshot service satisfies the conditions

required for eventual pursuit in the distributed pursuer evader tracking application.

2.5 Related work

2.5.1 Pursuer evader games using sensor network

In previous work, Schenato et al studied a pursuit evasion game application using

sensor networks. They considered a detailed system model with periodic time updates

and presented models of vehicle dynamics and uncertainty in track information. Sen-

sor network measurements are assumed to be fused at local stations to produce track

information. Evader assignment and pursuer control strategy is calculated at the base

station and then communicated to the pursuer agents. Network effects in communi-

cating this information to the pursuer agents and communicating pursuer locations

back to the base station are not considered. Within this framework, they derived a

series of algorithms to coordinate the pursuers so as to minimize the time-to-capture

of all evaders.

In this chapter, we have concentrated on the formulation of eventual pursuit con-

trol strategies despite network effects. We assume target track information has been

established through local fusion of sensor data. This track information is communi-

cated through the multi-hop wireless network infrastructure to pursuer agent, which

calculates a pursuit strategy based on evader and its own state.

2.5.2 Snapshot services

Communicating periodic global state snapshots is a well studied problem in dis-

tributed systems [17] and consistency, timeliness and reliability have been the main

48

design considerations in those studies. But efficiency becomes essential when con-

sidering periodic snapshots for resource constrained wireless sensor networks. To the

best of our knowledge algorithms for delivering periodic snapshots across a wireless

sensor network have not been studied before.

A common approach to achieving compression for efficiency is to exploit the tem-

poral and spatial correlation of data being shared. For example, in [66], the authors

propose a framework for a one time all-to-all broadcast of sensor data assuming the

data is spatially correlated. There has also been work [20] on compression mecha-

nisms for correlated sensor data sent to a central base station. Instead, in this paper

we do not require data to be correlated. At the same time, our algorithms can be

used in conjunction with other forms of compression.

Fractionally cascaded information [22] is a form of distance sensitive resolution

that is widely used in computational geometry community for speeding up data struc-

tures. Each node stores an ordered list of keys, shares a well distributed sample of

that list with its neighbors, a sample of that sample with its two hop neighbors, and

so on. Recently, fractional cascading has been used for sensor networks as an efficient

storage mechanism [28, 65]. Data is first stored at multiple resolutions across the

network, which is then used to efficiently answer aggregate queries about a range of

locations without exploring the entire area. In contrast, we have considered a model

where information is generated and consumed on an ongoing basis. Accordingly we

describe push based services that regularly deliver to subscribers snapshots of the net-

work in a pipelined manner. By providing snapshots with not just distance sensitive

-resolution but also -latency and -rate, we achieve compression and thereby efficiency.

49

At the same time these services can be used in range based querying as well as in

several other control applications.

An algorithm for creating the multi-resolution data structure based on probabilis-

tic gossip mechanism has been discussed in [65]. In [65], the algorithm described is

for a one shot dispersion and proceeds in stages while our services are for a model

where information is consumed on an ongoing basis and accordingly we describe a

pipelined implementation that is based on scheduling. In [65], the aggregation oper-

ations are duplicate insensitive and global time synchronization is assumed while we

do not require either of these properties. Our communication costs and latency are

lower than those in [65] and we also describe services that additionally have distance

sensitive rate properties. But we note that while we assume hierarchical clustering in

our solutions, the algorithm in [65] does not.

The idea of distance sensitive rate has also arisen in other contexts. Fisheye state

routing is a proactive routing protocol [62] that reduces the frequency of topology

updates to distant parts of the network. The spatial gossip protocol [40] is one in

which each node in a peer-to-peer network chooses to communicate to another node

with a probability that decreases polynomially with the distance between the pair.

Recently algorithms for bulk data collection in sensor networks have been pro-

posed. In [41] data is collected from one node at a time, while [58] performs con-

current, pipelined ex-filtration of data using TDMA schedules. In [49], the authors

describe TDMA based algorithms optimized for convergecast. Our algorithms can

be specialized for the case of bulk convergecast and we additionally emphasize on

efficiency using distance sensitive properties.

50

2.6 Summary

In this chapter, we considered a pursuer evader tracking scenario where multiple

pursuers and evaders exist in a bounded region. We presented a pursuer control

strategy and associated network support that guarantees an eventual catch despite

network effects. We showed that error, latency and update periods that decrease

linearly with distance are sufficient to achieve eventual catch in pursuer tracking

application.

We then designed wireless sensor network algorithms that periodically deliver

snapshots to all nodes in the network, which satisfy the above properties. Our algo-

rithms are easily adapted to allow snapshots to be delivered only to a subset of nodes

as opposed to all nodes. They are memory efficient, requiring only O(3f ∗log(N1f )∗m)

bits per node. They are readily realized in networks with irregular density, networks

with arbitrary sized holes, imperfect clustering, and non unit disk radios. We quan-

tify the maximum rate at which information can be generated at each node so that

snapshots are periodically delivered across the network, the algorithms can of course

be operated at lower rates than these. For our services, global time synchronization

is not required; a local notion of time however is needed to ensure fair scheduling of

transmission of nodes.

We have shown how our snapshot service can be used for the distributed pursuer

evader tracking application described above. In future, we would like to explore the

possibility of using this snapshot service in the context of other applications such

as control of distributed parameter systems and process control systems, in which

multiple distributed controllers require the state of the system periodically. We also

expect to implement our snapshot algorithms in the context of applications such

51

as pursuer evader tracking and vibration control, and study their performance and

tradeoffs more exhaustively in the future.

52

CHAPTER 3

DISTRIBUTED PURSUER EVADER TRACKINGAPPLICATION WITH OPTIMAL CATCH

In this chapter, we extend the pursuer evader tracking application to additionally

require that the evaders be caught in an optimal manner. Specifically, we consider a

pursuit-evasion game called “asset protection game” where pursuers try to protect a

linear target by intercepting the evaders as far as possible from the target. This game

structure has practical applications in real world applications of border and pipeline

protection and the techniques used to design this application can be generalized to a

wide variety of pursuit-evader games.

We assume target track information has been established through local fusion of

sensor data. This track information is communicated through the multi-hop wireless

network infrastructure to pursuer agent, which calculates optimal pursuit strategy

based on evader and its own state. But as the network is scaled in size, high through-

put rates for all pursuers cannot be sustained at all times, which necessitates a network

communication strategy that adapts to pursuer information requirements.

In this chapter, we first concentrate on the formulation of optimal pursuit control

strategies despite network effects. We adapt ideas from theory of differential games

to networked games in the presence of non-periodic track updates, message loss and

53

delays to derive optimal strategies, bounds on their information requirements and the

scaling properties of these bounds. In summary, we show (i) pursuer agents should

dictate the information refresh rate based on the requirements of the pursuit strategy,

and (ii) network delays and update periods should scale linearly with the pursuer-

evader distance to guarantee the existence of optimal min-max pursuit strategies

leading to Nash equilibrium.

We then describe an energy-efficient, fault-tolerant and hierarchy-free WSN ser-

vice, Trail, that supports the optimal pursuit requirements for the asset protection

game. The specification of Trail is to return the location of a particular object in

response to an in-network subscriber issuing a find query regarding that object. Trail

has a find cost that is linear (O(df)) in terms of the distance (df) of the subscriber

from the object. To this end, Trail maintains a tracking data structure by propa-

gating mobile object information only locally, and satisfying the distance sensitivity

requirement for the track updates. The amortized cost of updating a track when an

object moves a distance dm is O(dm ∗ log(dm)).

The rest of this chapter is organized as follows. First, in Section 3.1 we introduce

the game model. In Section 3.2, we review the optimal min-max strategies for the

pursuer and the evader under network communication constraints, and state lower

bounds on network performance requirements. Next, in Section 3.3, we present a

network service Trail that meets the required scalable information characteristics. In

Section 3.4, we describe the implementation of Trail on a 105 node Mica2 network,

that is used to support the asset protection game described above and use this im-

plementation to also study the performance of Trail. In Section 3.5 we discuss work

related to Trail. In Section 3.6, we present a summary.

54

3.1 Game Model

We consider a game between two players: a single pursuer and a single evader.

(For many n pursuer – n evader games the min-max solution can be reduced to n

two player games, by first solving the combinatorial problem of optimal pairing using

the value function of the two player game). The game state is given by the two

dimensional coordinates of the pursuer and evader x = xp, yp, xe, ye. Each player

travels at constant speed vp and ve and controls the direction of its motion, denoted

by θp and θe.

Linear asset

yp(T)

A(x

e(0),y

e(0))

Evader

B(x

p(0),y

p(0))

PursuerCInception Point

X

Y

Figure 3.1: The pursuer and evader game

We assume that there are no obstacles in the environment to constrain the move-

ment of the players. The players employ feedback strategies (up(x(t)), ue(x(t))) which

determine their direction of motion given the current state. The linear asset is as-

sumed to be infinitely long. With this assumption, the state space can be reduced to

three dimensions by defining relative coordinates, xr = xe −xp and yr = ye − yp. The

55

state vector x evolves according to:

x =∂

∂t

xr

yr

yp

= f(x, θp, θe) =

ve cos(θe) − vp cos(θp)ve sin(θe) − vp sin(θp)

vp sin(θp)

A catch is said to happen when x2r + y2

r < r2, where r is the catch radius. In the

following, we consider the limiting case of r → 0.

Starting from the initial condition x0, if the control strategies up(x), ue(x) satisfy

the catch condition at time T then the payoff is given by J (up, ue, x0) = yp(T ). The

game is zero-sum, so the pursuer’s goal is to maximize J whereas the evader’s goal

is to minimize J .

Min-max optimal feedback strategies u∗p(x), u∗

e(x) are defined by the saddle con-

dition:

Jup(up, u

∗e, x0) ≤ J (u∗

p, u∗e, x0) ≤ Jue

(u∗p, ue, x0) (3.1)

We also note that the min-max optimal strategy pair

u∗p(x), u∗

e(x) is also the Nash equilibrium [59] for this zero-sum game, where none

of the players have an incentive to change its strategy unilaterally given the rival is

maintaining its strategy choice.

3.2 Conditions for optimal interception

In this section, we first state the optimal min-max strategy when information

about evader is received continuously and instantaneously by the pursuer. We then

characterize the conditions on the maximum rate at which this information can be

received and also the maximum latency that be involved, in order to maintain the

optimality of pursuit strategy.

56

3.2.1 Optimal pursuit under perfect information

Given the current location of the evader and pursuer, the set of points that the

evader can reach before the pursuer is given by the well known Appolonius circle. The

min-max optimal strategies for the pursuer and evader is to directly to the boundary

point of the circle that is closest to the target. We state this formally in the following

theorem, the proof of which can be found in [15].

Theorem 3.2.1. If the ratio of the pursuer speed vp to the the evader speed ve α is

larger than 1, then the min-max optimal strategy for the evader and pursuers is given

by:

θe(x) = tan−1(tan γ + α√

1 + (tan γ)2) (3.2)

θp(x) = tan−1(tan γ +

√

1 + (tan γ)2

α) (3.3)

where γ = tan−1( yr

xr).

V (x) = yp +α2yr + α

√

y2r + x2

r

α2 − 1(3.4)

At each time instant t, the pursuer will calculate the best location (x′, y′)) that

the evader can reach:

x′ =xrα

2

α2 − 1+ xp (3.5)

y′ =α2yr + α

√

y2r + x2

r

α2 − 1+ yp (3.6)

then it will move towards that location.

We illustrate the performance of the optimal strategy using a simulation. The

results are given in Figure 3.2. The solid line shows the pursuer-evader trajectories

57

when both employ min-max optimal strategies. The dashed lines show the case when

evader uses non-optimal straight line strategies. We observe that min-max optimal

pursuit strategy catches non-optimal evaders at a larger distance to the target.

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

350 α=1.3

Pursuer1Evader1Pursuer2Evader2Pursuer3Evader3

Linear Asset

Figure 3.2: The P-E trajectory under perfect information

3.2.2 Sampling rate requirements of the optimal pursuit strat-

egy

In the previous subsection, we assumed that the global state is available to the

pursuer at all times. This is an unrealistic assumption for a sensor network imple-

mentation where the information can be provided only at discrete time intervals. In

this section, we characterize the sampling rate requirements of the optimal strategy

and show that it is inversely proportional to the relative distance between the pur-

suer and evader. we use the min-max solution concept to formulate a robust pursuit

strategy that will perform satisfactorily irrespective of evader motion. To design for

worst possible case of evader motion, we assume the pursuer has perfect information

about the location of the evader and the sampling period. The sampling period is

then chosen such that the evader does not benefit from switching from the optimal

58

direction given in Theorem. 3.2.1, although the evader’s deviation will be detected by

the pursuer after the sampling period interval.

Theorem 3.2.2. The evader does not deviate from its min-max equilibrium strategy

if and only if the distance moved by the pursuer before getting the next sample of state

information satisfies:

vpTsample <

√

α2(xr)2 + (α(yr) +√

(xr)2 + (yr)2)2

α(3.7)

Equivalently, the pursuer can move up to (α2−1)α2 of the total distance to the predicted

evader location before sampling the global state without loss of optimality.

The proof of this result can be found in [15]. The result is particularly important

for sensor network implementations using resource constrained nodes, because it in-

forms how the information data rate can be reduced based on the state of the game

so as to conserve the energy and bandwidth resources of the network.

We extend the previous result to derive the following scaling property of the

sampling period Tsample with respect to the distance dpe between the pursuer and

evader:

Theorem 3.2.3. Optimal pursuit-evasion strategies of the perfect information game

also yield Nash equilibrium of the game with discrete time updates if:

Tsamp(dpe) ≤α − 1

αvp

dpe

In other words, the sampling period should decrease proportionally with decreasing

distance between evader and pursuer to guarantee that the evader does not have an

incentive to deviate from its strategy to move directly to the predicted intercept point.

59

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

350 α=1.3

Pursuer1Evader1Pursuer2Evader2Pursuer3Evader3

Evader

Pursuer

Linear Asset

Figure 3.3: The P-E trajectory when using the Tsamp update

We illustrate the performance of the reduced sample rate strategy using a simu-

lation. The results are given in Figure 3.3. The solid line shows the pursuer-evader

trajectories when both employ min-max optimal strategies, which is identical to the

continuous update case. The dashed lines show the case when evader uses non-optimal

straight line strategies. We observe that reduced sample rate pursuit strategy differs

from its continuous information behavior for these cases but still catches these non-

optimal evaders at a larger distance to the target.

3.2.3 Effect of Packet Delay

The evader location information needs to be routed from the local fusion center

to the pursuer through wireless multiple hop links. The multiple hop communication

imposes considerable delays on the evader state information. We assume the network

is time synchronized and the packets are time-stamped at the source so that the

pursuer will able to calculate the delay of the packets it received. To derive a robust

pursuit strategy we design for the worst possible evader motion, by assuming the

evader will have perfect information about the pursuer location. Therefore at time

60

increment t evader have access to state information [xp(t), yp(t), xe(t), ye(t)] and the

pursuer have access to state information [xp(t), yp(t), xe(t − ∆t), ye(t − ∆t)]. Then

consider the following strategies:

Evader Strategy ue: The evader uses the current location information for the pursuer

to calculate the optimal direction as given in Theorem 3.2.1.

Pursuer Strategy up: The pursuer estimates the worst case location (xe(t), ye(t)) of

the evader by considering all the points that the evader can reach at ∆t and choosing

the one that yields the lowest game value V (xp(t), yp(t), xe(t), ye(t))

Theorem 3.2.4. The strategies up and ue are a Nash equilibrium of the pursuer-

evader game with packet delays if the delay at each point is bounded by:

∆t <α − 1

αvp

dpe(t − ∆t)

where dpe(t − ∆t) is the pursuer-evader distance at the time of packet transmission.

Linear asset

A(x

p(t),y

p(t))

B’(x

e(t−∆ t),y

e(t−∆ t))

B(x

e(t),y

e(t))

C(x,y)

θ’

θ

Figure 3.4: Effect of packet delay

61

In other words, the delay should decrease proportionally with decreasing distance

between evader and pursuer to guarantee that the evader does not have an incentive

to deviate from its strategy to move directly to the predicted intercept point.

3.3 Trail: Network service for distributed tracking

The results of the previous section indicate the following requirements on the

network protocol responsible for communicating evader track information to the pur-

suer agents: (i) Pursuer should determine the information refresh rate based on the

requirements of the pursuit strategy, and (ii) Network delays should scale with the

pursuer-evader distance. In this section, we describe a middleware service called Trail

that is compatible with these requirements.

The specification of Trail is to return the location of a particular object in response

to an in-network subscriber issuing a find query regarding that object. Trail has a

find cost that is linear (O(df)) in terms of the distance (df) of the subscriber from the

object. To this end, Trail maintains a tracking data structure by propagating mobile

object information only locally, and satisfying the distance sensitivity requirement for

the track updates. The amortized cost of updating a track when an object moves a

distance dm is O(dm ∗ log(dm)).

A basic Trail protocol can be refined by tuning certain parameters, thus resulting

in a family of Trail protocols. Appropriate refinements of the basic Trail protocol

are well suited for different network sizes and find/update frequency settings: One

refinement is to tighten its tracks by progressively increasing the rate at which the

tracking structure is updated; while this results in updating a large part of the tracking

structure per unit move, which is for large networks still update distance sensitive, it

62

significantly lowers the find costs for objects at larger distances. Another refinement

increases the number of points along a track, i.e., progressively loosens the tracking

structure in order to decrease the find costs and be more find-centric when object

updates are less frequent or objects are static. Moreover, Trail scales well to networks

of all sizes. As network size decreases, Trail gradually eschews local explorations and

updates and thus increasingly centralizes update and find.

We evaluate the performance of Trail by simulations in a 90×90 sensor network and

experiments on 105 Mica2 nodes in the Kansei testbed[3]. This implementation has

been used to support a distributed intruder interceptor tracking application where the

goal of the interceptor is to catch the intruders as far away from an asset as possible.

Overview of solution: Trail maintains tracks from each object to only one

terminating point, namely, the center of the network C; these tracks are almost

straight to the center, with a stretch factor of at most 1.2 times the distance to C.

Note that if the track to an object P is required to be always a straight line from C

to the current location of P resulting in a stretch factor equal to 1, then every time

the object moves, the track has to be updated starting from C, which would not be

a distance sensitive approach. Therefore, we form the track as a set of trail segments

and update only a portion of the structure depending upon the distance moved. Thus

given a terminating point Trail maintains a track with lengths from the terminating

point that is almost close to minimum. This is important because longer tracks have

a higher cost of initialization and given that network nodes may fail due to energy

depletion or hardware faults, longer tracks increase the probability of a failed node

along a track as well as increase the cost of detecting and correcting failures in the

track.

63

Given the track to an object, a find operation explores along circles of exponen-

tially increasing radii until the track is intersected and then follows the track to the

current location of the object. The tracks maintained in Trail only contain pointers

to the current location of an object and not the state information of the object. Pub-

lishing the current state (namely location) of the object along all points in track will

violate distance sensitivity of updates because every move of the object will result in

updating the entire track. Following the trail of an object from any location leads to

the current location of the object which contains the state of the object. Yet Trail is

distance sensitive in terms of the find in the sense that the total cost of reaching the

track for an object, and following the track to reach the object is proportional to the

distance of the finder from the object. Note that a find explores along circles until

a radii that is at most half the distance of the finder to C and then searches at C

where the track is certain to be found. Thus C serves as a worst case landmark for

finding objects in the network.

In our solution, we make some design decisions like choosing a single point to

terminate tracks from all points in the network and avoiding hierarchy in maintaining

the tracks. In Subsection 3.3.6, we analyze these aspects of our solution and compare

them with other possible approaches.

In Subsection 3.3.1, we formally state the system assumptions for Trail and the

specification. In Subsection 3.3.2, we design the basic Trail protocol for a 2-d real

plane. Then, in Subsection 3.3.3, we present an implementation of the basic Trail

protocol for a 2-d sensor network. In Subsection 3.3.4, we discuss refinements of the

basic Trail protocol. In Subsection 3.3.6, we analyze some design decisions made in

64

our solution and compare them with other possible approaches. In Subsection 3.3.7,

we present results of our performance evaluation in simulation.

3.3.1 System Model and Specification

The system consists of a set of mobile objects, and a network of static nodes that

each consist of a sensing component and a radio component. Tracking applications

execute on the mobile objects and use the sensor network to track desired mobile

objects. Object detection and association services execute on the nodes, as does the

desired Trail network tracking service.

The object detection and association service assigns a unique id, P , to every object

detected by nodes in the network and stores the state of P at the node j that is

closest to the object P . This node is called the agent for P and can be regarded

as the node where P resides. The problem of detecting objects in the network and

uniquely associating them with previous detections is thus orthogonal to the tracking

service that we discuss in this paper. Detection and association services can be

implemented in a centralized [69] or distributed [67] fashion; the latter approach would

suit integration with the tracking service that we discuss in this paper. Detection and

association could also be enabled by the objects being cooperative, for example, being

tagged with RFID and announcing their identifier. We assume that the underlying

detections and associations are always correct.

Trail Network Service: Trail maintains an in-network tracking structure, trailP ,

for every object P . Trail supports two functions: find(P, Q), that returns the state of

the object P , including its location at the current location of the object Q issuing the

65

query and move(P, p’, p) that updates the tracking structure when object P moves

from location p′ to location p.

Definition 10 (find(P, Q) Cost). The cost of the find(P, Q) function is the total

communication cost of reaching the current location of P starting from the current

location of Q.

Definition 11 (move(P, p’, p) Cost). The cost of the move(P, p′, p) function is the

total communication cost of updating trailP to the new location p and deleting the

tracking structure to the old location p′.

We note that our network service does not assume knowledge of the motion model

of objects being tracked, in contrast to some query services [53], and as such the scope

of every query in our case is the entire network as opposed to a certain locality. Nor

does it assume a bound on the number of querying objects in the network or any

synchrony between concurrent queries.

Network Model: To simplify our presentation, we first describe Trail in a 2-d

real continuous plane. We then refine the Trail protocol to suitably implement in a

random connected deployment of a wireless sensor network. In this model, we impose

a virtual grid on the random deployment and snap each node to its nearest grid

location (x, y). Each node is aware of this location. We refer to unit distance as the

one hop communication distance. dist(i, j) now stands for distance between nodes i

and j in these units. We describe this model in more detail and the implementation

of Trail in this discrete model in Section 4.

Fault Model: In the wireless sensor network, we assume that nodes can fail due to

energy depletion or hardware faults or there could be insufficient density at certain

66

regions, thus leading to holes in the network. However, we assume that the network

may not be partitioned; there exists a path between every pair of nodes in the network.

3.3.2 Trail on a 2-d Plane

In this section, we use geometric ideas to design Trail for a bounded 2-d real plane.

Let C denote the center of this bounded plane.

Tracking Data Structure

We maintain a tracking data structure for each object in the plane. Let P be an object

being tracked, and p denote its location on the plane. We denote the tracking data

structure for object P as trailP . Before we formally define this tracking structure,

we give a brief overview.

Overview: If trailP is defined as a straight line from C to P , then every time

the object moves, trailP has to be updated starting from C. This would not be a

distance sensitive approach. Hence we form trailP as a set of trail segments and

update only a portion of the structure depending upon the distance moved. The

number of trail segments in trailP increases as dist(p, C) increases. The end points of

each trail segment serve as marker points to update the tracking structure when an

object moves. The point from where the update is started depends on the distance

moved. Only, when P moves a sufficiently large distance, trailP is updated all the

way from C. We now formally define trailP .

Definition 12 (trailP ). The tracking data structure for object P , denoted by trailP ,

for dist(p, C) ≥ 1 is a path obtained by connecting any sequence of points (C, Nmx, ..., Nk, ..., N1, p)

by line segments, where mx ≥ 1, and there exist auxiliary points c1..cmx that satisfy

67

the properties (P1) to (P3) below. mx is defined as d(log2(dist(C, po)))e − 1, where

po is the location of p when trailP was initialized or updated starting from C.

For brevity, let Nk be the level k vertex in trailP ; let the level k trail segment

in trailP be the segment between Nk and Nk−1 ; let Seg(x, y) be any line segment

between points x and y in the plane.

• (P1): dist(ck, Nk) = 2k, (mx ≥ k ≥ 1).

• (P2): Nk−1, (mx ≥ k ≥ 1), lies on Seg(Nk, ck−1); Nmx lies on Seg(C, cmx).

• (P3): dist(p, ck) < 2k−b, (mx ≥ k ≥ 1) and b ≥ 1 is a constant.

If (dist(p, C) = 0), trailP is C; and if (0 ≤ dist(p, C) < 1), trailP is Seg(C, p).

(a) Initial (b) P moves 3 unitsfrom c3

(c) P moves backtowards c3

Figure 3.5: Examples of Trail to an Object P

68

Observations about trailP : From the definition of trailP , we note that the

auxiliary points c1..cmx are used to mark vertices N1..Nmx of trailP . P1 and P2

describe the relation between the auxiliary points and the vertices of trailP . Given

trailP , points c1..cmx are uniquely determined using P1 and P2. Similarly given p and

c1, ..cmx, trailP is uniquely determined. These properties are stated in the following

Lemmas.

Lemma 3.3.1. Given trailP , points c1..cmx are uniquely determined.

Proof. Extend Seg(C, Nmx) of trailP by a distance of 2mx to obtain cmx. Similarly

extend Seg(Nk, Nk−1) by 2k−1 to obtain ck−1 for 0 < k ≤ mx. Thus using properties

P1 and P2 of trailp, points c1..cmx are uniquely determined given C, Nmx, .., N1, p of

trailP .

Lemma 3.3.2. Given c1, ...cmx and p, trailP is uniquely determined.

Proof. Nmx lies on Seg(C, cmx) such that dist(cmx, Nmx) = 2mx. Similarly Nk−1

lies on Seg(Nk, ck−1) such that dist(ck−1, Nk−1) = 2k−1 for 1 < k ≤ mx. Thus

C, Nmx, .., N1, p of trailP are uniquely determined.

By property P3, the maximum separation between p and any auxiliary point ck

decreases exponentially as k decreases from mx to 1. When an object moves a certain

distance away from its current location, trailP has to be updated from the smallest

index k such that property P3 holds at all levels. By changing parameter b in property

P3, we can tune the rate at which the tracking structure is updated. We discuss these

refinements in Section 3.3.4.

Note from the definition of trailP that mx is defined as d(log2(dist(C, po)))e − 1

where po was the location of the object when trailP was either created or updated from

69

C. The value of mx which denotes the number of trail segments in trailP , depends

on the distance of P from C. When trailP is first created, c1, ..., cmx are initialized

to location po, the number of levels mx is initialized to d(log2(dist(C, po)))e − 1 and

trailP is a straight line. The value of mx is updated when trailP has moved a sufficient

distance to warrant an update of trailP all the way from C. The update (and create)

procedure for trailP is described in more detail in the following subsection.

We now show 3 examples of the tracking structure in Fig. 1. In this figure, b = 1.

Fig. 3.5(a) shows trailP when c3, ..c1 are collocated. When P moves away from this

location, trailP is updated and Fig. 3.5(b) shows an example of trailP where c2, c1

are displaced from c3. In Fig. 3.5(b), dist(c3, c2) = 2 units, dist(c2, c1) = 1 unit, and

p and c1 are collocated. Moreover, N3 lies on Seg(C, c3), N2 lies on Seg(N3, c2) and

so on. In Fig. 3.5(c) we show an example of a zigzag trail to an object P , when P

moves away from c3 and then moves back in the opposite direction.

Updating the trail

We now describe a procedure to update the tracking structure when object P

moves from location p′ to p such that the properties of the tracking structure are

maintained and the cost of update is distance sensitive.

Overview: When an object moves distance d away, we find the minimal index m,

along trailP such that dist(p, cj) < 2j−b for all j such that mx ≥ j ≥ m and trailP is

updated starting from Nm. In order to update trailP starting from Nm, we find new

vertices Nm−1, ..., N1 and a new set of auxiliary points cm−1, ..., c1. Let N ′m−1, ..., N

′1

and c′m−1, ..., c′1 denote the old vertices and old auxiliary points respectively. Starting

from Nm, we follow a recursive procedure to update trailP . This procedure is stated

below:

70

Figure 3.6: Updating trailP

Update Algorithm:

1. Let m be the minimal index on the trail such that dist(p, cj) < 2j−b for all j

such that mx ≥ j ≥ m.

2. k = m

3. while k > 1

(a) ck−1 = p; Now obtain Nk−1 using property P2 as follows: the point on

segment Nk, ck−1, that is 2k−1 away from ck−1.

(b) k = k − 1

If no indices exist such that dist(p, cj) < 2j−b, then the trail is created starting from

C. This could happen if the object is new or if the object has moved a sufficiently large

distance from its original position. In this case, mx is set to (dlog2(dist(C, p))e) − 1.

cmx is set to p. Nmx is marked on Seg(C, p) at distance 2mx from cmx. Step 2 is

executed with k = mx.

71

Fig. 3.6 illustrates an update operation, when b = 1. In Fig. 3.6a, dist(p, p′) is 2

units. Hence update starts at N3. Initially c3, c′2, c′1 are at p′. We use the update

algorithm to determine new c2, c1 and thereby the new N2, N1. Using step (3a) of the

update algorithm, the new c2 and c1 lie at p. The vertex N2 then lies on Seg(N3, c2)

and N1 lies on Seg(N2, c1). In Fig. 3.6b, P moves further one unit. Hence update

now starts at N2. Using step (3a) of the update algorithm, the new c1 lies at p and

N1 lies on Seg(N2, c1).

Note: The Trail update algorithm described above is for a continuous 2-d plane.

In this algorithm, we have stated the minimum level from which trailP is updated

and described how the vertices and auxiliary points are updated starting from this

level. In Section 3.3.3, we describe Trail protocol on a wireless sensor network. A

formal specification of the algorithm, in the form of guarded command actions, is also

provided in Section 3.3.3.

Lemma 3.3.3. The update algorithm for Trail yields a path that satisfies trailP .

Proof. 1. Let m be the index at which update starts. By the condition in step 1,

dist(cj, p) < 2j−b for all mx ≥ j ≥ m. Now, for m > j ≥ 1, cj = p. Therefore

for m > j ≥ 1, dist(cj, p) < 2j−b. Thus property P3 is satisfied.

2. Properties P2 and P1 are satisfied because m ≥ k > 1, we obtain Nk−1 as the

point on Seg(Nk, ck−1), that is 2k−1 away from ck−1.

3. mx is defined for trailP , when trailP is created or updated starting from C.

When mx is (re)defined for trailP , cmx is the position of the object and mx is

set to (dlog2(dist(C, p))e) − 1.

72

Definition 13 (Trail Stretch Factor). Given trailP to an object p, we define the trail

stretch factor for any point x on trailP as the ratio of the length along trailP from x

to p, to the Euclidean distance dist(x, p).

Lemma 3.3.4. The maximum Trail Stretch Factor for any point along trailP , denoted

as TSp is sec(α) ∗ sec(α2) where α = arcsin( 1

2b ).

(a) Max Angle in Trail (b) Analyzing Stretch

Figure 3.7: Analyzing Trail Stretch Factor

Proof. We prove Lemma 3.3.4 by using the following steps.

(Maximum angle (∠pNkck)) Let the maximum angle formed by p and ck at Nk

in trailP for (mx ≥ k ≥ 1) be denoted as α. Refer Fig. 3.7(a). Recall from properties

of trailP that dist(Nk, ck) = 2k and dist(p, ck) < 2k−b. Note that ∠pNkck is maximum

when Seg(Nk, p) is tangent to a circle of radius 2k−b and center ck. Therefore we have

the following condition.

73

α < arcsin(1

2b) (3.8)

(Maximum stretch at a given level k) Let x be any point on trailP which lies

on Seg(Nk+1, Nk). Refer Fig. 3.7(b). We note the following equation based on the

geometry of Fig. 3.7(b).

(dist(x, Nk) + dist(Nk, p))

dist(x, p)=

(sin(θ) + sin(φ))

sin(θ + φ)(3.9)

Also note that (θ + φ) = ∠pNkck. Using this, we get Eq. 3.10

(θ + φ) ≤ α (3.10)

Let f(θ, φ) denote the following function.

f(θ, φ) =(sin(θ) + sin(φ))

sin(θ + φ)(3.11)

It can be shown that f(θ, φ), where θ > 0, φ > 0 and θ + φ ≤ α is maximum

when θ = φ = α2. We state this as Proposition A.0.1 and prove it in Appendix A.

Substituting (θ = φ) in Eq. 3.9, we get the following condition.

(dist(x, Nk) + dist(Nk, p))

dist(x, p)≤ sec(

α

2) (3.12)

Thus, we see that at a single level k the maximum stretch for trailP occurs when

∠pNkck is α for a point x on Seg(Nk+1, Nk) such that ∠pxNk = ∠xpNk. Since

∠pxNk = ∠xpNk when the maximum occurs, we also get the following equation.

(dist(x, Nk))

dist(x, p)=

(dist(Nk, p))

dist(x, p)=

sec(α2)

2(3.13)

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(Maximum trail stretch factor from vertex Nk to p) In order to find the

maximum stretch factor over multiple levels, we consider trailP to be split at vertices

Nk to N2 in such a way that the stretch is maximized at each level. Thus we let

∠pNjcj = α and ∠pNjNj−1 = ∠NjpNj−1 for all k ≥ j > 1. In Fig. 3.8, we show one

such trail segment, Seg(Nj, Nj−1) of trailP .

Figure 3.8: Analyzing Trail Stretch

Using Eq. 3.13, we get the following equations:

dist(Nj−1, p)

dist(Nj, p)=

sec(α2)

2∀j : k ≥ j > 1 (3.14)

dist(Nj, Nj−1)

dist(Nj, p)=

sec(α2)

2∀j : k ≥ j > 1 (3.15)

When the above configuration is repeated at all levels of trailP , we determine the

ratio of the lengths of two successive trail segments, Seg(Nj−1, Nj−2) and Seg(Nj, Nj−1).

Using Eq. 3.14 and Eq. 3.15, we get the following equation.

75

dist(Nj−1, Nj−2)

dist(Nj , Nj−1)=

sec(α2)

2∀j : k ≥ j > 2 (3.16)

Let Lk denote the length along trailP from vertex Nk. Using Eq. 3.16, we get:

Lk = dist(N2, N1) ∗j=(k−2)∑

j=0

(2 ∗ cos(α

2)j) + dist(N1, p) (3.17)

Let Rk denote the trail stretch factor from Nk.

Rk =Lk

dist(Nk, p)(3.18)

Upon simplification using Eq. 3.14, Eq. 3.15 and Eq. 3.17, we get:

Rk =1

2 ∗ cos(α2) − 1

+1

(2 ∗ cos(α2))k−1

∗ (1 − 1

2 ∗ cos(α2) − 1

) (3.19)

Since, α < π6

for b ≥ 1, 0 < 2 ∗ cos(α2) − 1 ≤ 1. Therefore we get:

Rk ≤ 1

2 ∗ cos(α2) − 1

(3.20)

Since, cos(α) = 2 ∗ cos2(α2) − 1, cos(α

2) ≤ 1 and 0 < 2 ∗ cos(α

2) − 1 ≤ 1, we get:

Rk ≤ 1

cos(α)(3.21)

(Maximum trail stretch factor from any point in trailP to p) Let x be any

point on trailP which lies on a level k + 1 segment, i.e Seg(Nk+1, Nk), but is not a

vertex point. Let Lxp denote the length along trailP from x to p. Using Eq. 3.21 and

Eq. 3.12, we have the following inequalities:

76

Lxp ≤ dist(x, Nk) + dist(Nk, p) ∗ sec(α)

≤ (dist(x, Nk) + dist(Nk, p)) ∗ sec(α)

≤ sec(α

2) ∗ sec(α) ∗ dist(x, p)

Thus we have proved that the maximum Trail Stretch Factor for any point along

trailP , denoted as TSp is sec(α) ∗ sec(α2) where α = arcsin( 1

2b ).

Lemma 3.3.5. The length of trailP for an object P starting from a level k(0 < k ≤

mx) vertex, denoted as Lk is bounded by (2k + 2k−b) ∗ TSp.

Proof Sketch: dist(ck, p) < 2k−b. Therefore, dist(Nk, p) < 2k + 2k−b. Then using

Lemma 3.3.4, the result follows.

Theorem 3.3.6. The upper bound on the amortized cost of updating trailP when

object P moves distance dm(dm ≥ 1) is 4 ∗ (2b + 1) ∗ TSp ∗ dm ∗ log(dm).

Proof. Note that in update whenever trailP is updated starting at the level k vertex,

we set ck−1 = p. P can now move a distance of 2k−1−b before another update starting

at the level k vertex. Thus, between any two successive updates starting from a level

k vertex, the object must have moved at least a distance of 2k−1−b. The total cost

to create a new path and delete the old path starting from a level k vertex costs at

most 2 ∗ Lk.

When an object moves a total distance dm where dm ≥ 1, it could involve multiple

updates at smaller distances. The object could be detected at multiple instances over

this distance dm. Therefore we calculate the upper bound on the amortized cost of

update when the object moves distance dm. We consider the minimum distance to

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trigger an update to be 1 unit. Note that between any two successive updates starting

from a level k vertex, the object must have moved at least a distance of 2k−1−b. Thus

over a distance dm, update can start at level (b + 1) vertex at most dm times, update

can start at level (b + 2) vertex can at most bdm/2c times, and so on. The update

can start at level (blog2(dm)c+ b+1) vertex at most once. Adding the total cost, the

result follows.

Remark on update cost: We note the following points about the update cost

characterized in Theorem 3.3.6. (1) First of all, over a distance of dm, the object can

be detected in multiple instances and consequently multiple update of the track can

result over a distance dm, but the total cost of all the updates over a distance dm is

O(dm ∗ log(dm)). We consider the minimum distance to trigger an update to be 1

unit. We add the total cost resulting from each update and the sum of the cost of

all these possible updates results in an upper bound on the amortized cost, stated in

Theorem 3.3.6. (2) Secondly, it is only the amortized cost of update which is of the

order of O(dm ∗ log(dm)) and distance sensitive. During a move of distance dm by any

object P , there could be updates triggered over a smaller distance dms (for instance,

dms = 1), that result in trailP being updated from a much larger level k >> 1. But

the update algorithm of Trail guarantees that whenever trailP is updated starting at

the level k vertex, P can now move a distance of 2k−1−b before another update starting

at the level k vertex. Thus, between any two successive updates starting from a level

k vertex, the object must have moved at least a distance of 2k−1−b. There can only

be updates at levels smaller than k between two updates at level k. Upon adding

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up all these update costs, we get the amortized update cost which is of the order of

O(dm ∗ log(dm)) and distance sensitive.

b Trail Stretch Update Cost1 1.2 14 ∗ dm ∗ logdm

2 1.05 21 ∗ dm ∗ logdm

> 3 Approaches 1 4 ∗ (2b + 1) ∗ dm ∗ logdm

Figure 3.9: Effect of b on Update cost

For illustration, we summarize the Trail Stretch factor and update costs for dif-

ferent values of b in Fig. 3.9. We explain the significance of the refinement of Trail

by varying b in Section 5.

Basic Find Algorithm

Given trailP exists for an object P , we now describe a basic find algorithm that

is initiated by object Q at point q on the plane. We use a basic ring search algorithm

to intersect trailP starting from Q in a distance sensitive manner. We then show

from the properties of the Trail tracking structure that starting from this intersection

point, the current location of P is reached in a distance sensitive manner.

Basic find Algorithm:

1. With center q, successively draw circles of radius 20, 21, ...2blog(dist(q,C))c−1, until

trailP is intersected.

2. If trailP is intersected, follow it to reach object P ; else follow trailP from C

(note that if object exists, trailP will start from C).

79

(a) find path (b) Farthest find point

Figure 3.10: Basic find algorithm in Trail

Theorem 3.3.7. The cost of finding an object P at point p from object Q at point q

is O(df) where df is dist(p, q).

Proof. Note that as q is distance df away from p, a circle of radius 2dlog(df )e will

intersect trailP . Hence the total length traveled along the circles before intersecting

trailP at point s is bounded by 2 ∗ π ∗ ∑dlog(df )ej=1 2j , i.e., 8 ∗ π ∗ df . The total cost of

connecting segments between the circles is bounded by 2 ∗ df .

Now, when the trail is intersected by the circle of radius 2dlog(df )e, the point s at

which the trail is intersected can be at most 3 ∗ df away from the object p. This is

illustrated in Fig. 3.10(b). In this figure, q is df + ∇ away from p. Hence the trail

can be missed by circle of radius 2log(df ). From Lemma 3.3.4, we have that distance

along the trail from s to p is at most 3 ∗ TSp ∗ df . Thus, the cost of finding an object

P at point p from object Q at point q is O(df) where df is dist(p, q).

Update cost when consisting of discrete jumps: Note that the update costs

characterized in Fig. 3.9 are for the continuous update case when updates are per-

formed after every unit distance of move. Now consider the case where a total move

80

of distance dm consists of multiple discrete jumps. In each jump, an object disap-

pears at a certain location and is later detected at another location at distance dx

away such that dx >> 1. In this case, we note that the cost of individual smaller

updates need not be added. However there is an additional cost of exploring to find

an existing trail because we do not assume memory of the previous location of an

object. (In the continuous setting where updates performed after every one unit of

move, we ignore the cost of finding an existing trailP as it will exist within a dis-

tance of 1 unit.) But recall from the find algorithm that when P moves distance dx

away, an existing trailP will be intersected at an upper bound cost of 8 ∗ π ∗ dx, i.e.

O(dx). Therefore the amortized update cost still remains O(dm ∗ log(dm)) over a total

distance dm consisting of such discrete jumps.

3.3.3 Implementing Trail in a WSN

In this section, we describe how to implement the Trail protocol in a WSN that

is a discrete plane, as opposed to a continuous plane as described in the previous

section. Trail can be implemented in any random deployment of a WSN aided by

some approximation for routing along a circle. We describe one such implementation

below.

System model: Consider any random deployment of nodes in the WSN. We impose

a virtual grid on this deployment and snap each node to its nearest grid location (x, y)

and assume that each node is aware of this location. We refer to unit distance as the

one hop communication distance. dist(i, j) now stands for distance between nodes

i and j in these units. The separation along the grid is less than or equal to the

unit distance. When the network is dense, the grid separation can be smaller. The

81

neighbors of a node are a set of all nodes within unit distance of the node. Thus when

the grid separation is unit distance, there are at most 4 neighbors for each node. We

also assume the existence of an underlying geographic routing protocol such as GPSR

[38], aided by an underlying neighborhood service that maintains a list of neighbors

at each node.

Note: We have implicitly assumed in the above model that each location on the grid

is mapped to a unique node. This can be achieved by decreasing the size of the grid

to be equal to the smallest separation between any 2 nodes in the network. However

this assumption is not necessary. In other words, all nodes need not be necessarily

assigned to some location on the grid. Nodes can take turns to play the role of a

given location. But for ease of exposition, we have abstracted away these possibilities

in our model.

Fault Model: We assume that nodes in the network can fail due to energy depletion

or hardware faults, or there could be insufficient density at certain regions, thus

leading to holes in the network. However, we assume that the network may not be

partitioned; there exists a path between every pair of nodes in the network. A node

may also transiently fail. But we assume that the failed nodes return in a clean or

null state without any knowledge of tracks previously passing through them. We do

not consider arbitrary state corruptions in the nodes.

When implementing on a WSN grid, Trail is affected by the following factors:(1)

discretization of points to nearest grid location; (2) Overhead of routing between any

two points on the grid; and (3) holes in the network. We discuss these issues in this

section.

82

Routing stretch factor: When using geographic routing to route on a grid, the

number of hops to communicate across a distance of d units will be more than d. We

measure this stretch in terms of the routing stretch factor, defined as the ratio of the

communication cost (number of transmissions) between any two grid locations, to

the Euclidean distance d between two grid locations. It can be shown that the upper

bound on the routing stretch factor for the WSN unit grid is√

2. If we consider the

grid to be of smaller separation than the communication range (denser grid), then

the routing stretch factor will decrease as any straight line will now be approximated

more closely when moving along the grid.

(a) find in a WSN grid (b) update in a WSN grid

Figure 3.11: Find and update algorithm in a WSN grid

Implementing Update on WSN Grid

Storage: For each object P in the network, trailP is maintained by parent / child

pointers at each node in the network. Starting from C, following the child pointers

for P will lead to the current location of P . Similarly, starting from p, following

the parent pointers at each node will lead to C. Some of the nodes along trailp are

marked as vertex nodes. Each node keeps memory of whether it is a vertex node.

83

Each vertex node at level m keeps memory of ci for all levels mx ≥ i ≥ m, which is

used to determine the smallest level at which an update should start. An array of

size log(N) is allocated at each node to store the auxiliary points, where the network

is of size N × N .

Protocol Trail at node j in NxN networkVar

j.childp : child pointer for object pj.prntp : parent pointerj.detectp : booleanj.levelp : level at which node j belongsj.vertexp : boolean indicating if j is a vertexj.mxp : maximum levelsj.cp : array [0..log(N)] of auxiliary pointsnh : auxiliary variable to store the next hop for any message

Figure 3.12: Trail: State at Node j for Object P

The state of node j for a given object P being tracked is shown in Fig. 3.12. The

actions involved in updating trailP (for b = 1) are shown in Fig. 3.13 and Fig. 3.14.

A description of these actions follows.

Actions: We use three types of messages in the update actions. The update actions

are initiated at a node where an object P is detected. Recall from Section 2 that this

detection is flagged by the underlying detection and association service at the node

that is closest to the location of P .

Initially, when an object is detected at a node j, it sends an explore message

(Action U1). The parameters of this message are the object id and the location of the

object. The explore message travels in around the square perimeters of side lengths,

20, 21, ...2blog(dist(q,C))c−1 until it meets trailP or else the explore message travels to C.

84

Note that if the object is updated continuously as it moves, then the explore message

will intersect the trail within a 1 hop distance (first level of search). The auxiliary or

thought variable nh is used to refer to the next hop of any message.

When an explore message is received at a node j (Action U2), if the node does

not belong on trailP , it simply forwards the message to the next hop of exploration.

If explore message intersects trailP , the message is forwarded along its parent pointer

until the level m vertex node where m is the minimal index such that dist(cm, p) <

2m−1 for all i such that mx ≥ i ≥ m. Starting from the level m node where update is

started, a new track is created by sending a grow message towards the current location

of P . The parameters of a grow message are the identifier of the object, the current

location of the object, the level w of the subsequent vertex, the maximum levels of

trailP and the array containing values of ci for all levels mx ≥ i ≥ m. Geographic

routing is used to route the message towards the current location. If the exploration

reaches C, the maximum levels of trailP are reset and a grow message is sent towards

the location of P .

Upon receiving a grow message (Action U3), node j checks to see if it is a vertex

node. The node closest to, but outside or on a circle of radius 2w around cw is marked

as Nw. This node copies the values of maximum levels in trailp, the level of node j in

trailP , and the values of ci for all mx ≥ i > w, from the grow message. Additionally,

the value of cw is set to current location of P . The child and parent pointers are

updated to the sender of the grow message and the next hop towards the current

location of P respectively. The grow message is then forwarded to the next hop using

geographic routing [38]. If node j is not a vertex node, it simply marks its level in

trailP , updates the parent and child pointers and forwards the message to the next

85

〈U1〉 :: ((j.detectp) ∧ (j.childp 6= j)) −→j.childp = j;nh = nexthop of exploration;send(j,nh) (explore(p, j));

[]〈U2〉 :: recvk,j(explore(p, cur)) −→

if (j == C)nh = nexthop towards cur;j.childp, j.mxp = nh, dlog(dist(C, cur))e − 1;send(j,nh) (grow(p, cur, j.mxp, j.mxp, j.cp));

else

if (¬j.child)nh = nexthop of exploration;send(j,nh) (explore(p, cur));

else

if ((j.vertexp) ∧ (∀s : (j.mxp ≥ s ≥ j.levelp) : (dist(j.cp[s], cur) < 2s−1)))send(j,j.childp) clear(p)

nh = nexthop towards cur;send(j,nh) grow(p, cur, j.levelp − 1, j.mxp, j.cp);

j.child = nh;else

send(j,j.prntp) (explore(p, cur));

fi

fi

fi

Figure 3.13: Trail: Update Actions (U1 and U2)

86

hop towards the current location of P . This procedure is then repeated at subsequent

nodes and the trail is updated. If j is the current location of the object, the grow

message is not propagated further. Fig. 3.11(b) shows how a trail is updated in the

grid model with the grid spacing set equal to the unit communication distance. The

vertex pointers N3, ...N1 are shown approximated on the boundary of the respective

circles.

Also, starting from the level m node where update is started, a clear message is

used to delete the old path. Upon receiving a clear message (Action U4), node j

forwards the message to its child and the state of j with respect to object P is reset.

The clear message is passed along child pointers of trailP until the node where object

P previously resided.

Implementing find on WSN Grid

We now describe how to implement the find algorithm in the WSN grid. As seen

in Section 3.3.2, during a find, exploration is performed using circles of increasing

radii around the finder. However, in the grid model, we approximate this procedure

and instead of exploring around a circle of radius r, we explore along a square of side

2 ∗ r. The perimeter of the square spans a distance 8 ∗ r instead of 2 ∗ π ∗ r. We

characterize the upper bound on the find cost in the following Lemma.

Lemma 3.3.8. The upper bound on the cost of finding an object P at point p from

object Q at point q is (32 + 3 ∗ sec(α) ∗ sec(α2) ∗

√2) ∗ df where df is dist(p, q) and

α = arcsin( 12b ).

Proof. In the WSN virtual grid, the total cost of exploring along squares up to level

2log(df )is given by 8 ∗∑dlogde

j=0 2j, i.e 32 ∗ df . Recall from the proof of Theorem 3.7 that

87

〈U3〉 :: recvk,j(grow(p, cur,w, x, c)) −→if (j == cur)

j.prntp, j.levelp, j.mxp = k,w, x;else

nh = nexthop towards cur;if (dist(j, cur) ≥ 2w)) ∧ (dist(j, nh) < 2w))

j.vertexp, j.levelp, j.childp, j.mxp = true,w, nh, x;Resetj.cp; j.cp = c; j.cp[j.levelp] = m;send(j,nh) grow(p, cur,w − 1, j.mxp, j.cp);

else

j.prntp, j.levelp = k,w;send(j,nh) grow(p, cur,w, x, c);

j.childp = nh;fi

fi

[]〈U4〉 :: recvk,j(clear(p)) −→

if ((j.childp 6= j))send(j,j.childp) (clear(p);

fi

j.childp, j.prntp, j.vertexp, j.cp, j.levelp = ⊥,⊥,⊥,⊥,⊥;

Figure 3.14: Trail: Update Actions (Actions U3 and U4)

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when the trail is intersected by the circle of radius 2dlog(df )e, the point s at which the

trail is intersected can be at most 3 ∗df away from the object p. The cost of reaching

p from the point of intersection with trailP is bounded by 3∗df ∗TSp ∗√

2 where TSp

is the maximum trail stretch factor possible for P . Note that there is an additional

stretch of√

2 because of routing along a grid. The result follows.

The actions for the find algorithm are shown in Fig. 3.15. Upon receiving a

find(p,q) message at node j (Action F1), if node j does not belong to trailP , then the

message is forwarded to the next hop of exploration or else the message is forwarded

to the child node. If the current location of P is reached, then a found message is

sent towards q. Upon receiving a found message (Action F2), the state of object P

is returned to q using geographic routing.

Fault-Tolerance

Due to energy depletion and faults, some nodes may fail leading to holes in the WSN.

Trail supports a graceful degradation in performance in the presence of node failures.

As the number of failures increase, there is only a proportional increase in find and

update costs as the tracking data structure and the find path get distorted. This is

especially good when there are a large number of small holes in the network that

are uniformly distributed across the network as has been the case in our experiments

with large scale wireless sensor networks [9]. We discuss the robustness of Trail under

three scenarios: during update, during find and maintaining an existing trail.

Tolerating node failures during update: A grow message is used to update a

trail starting at a level k node and is directed towards the center of circle k−1. In the

89

〈F1〉 :: recvk,j(find(p, q)) −→if (j.childp == ⊥)

nh = nexthop of exploration;sendj,nh (find(p, q)) ;

[](j.childp 6= j) ∧ (j.childp 6= ⊥)

sendj,j.childp(find(p, q)) ;

[](j.childp = j);

nh = nexthop towards p;sendj,nh (found(p, q)) ;

fi

[]〈F2〉 :: recvk,j(found(p, q));−→

if (j 6= q)nh = nexthop towards p;sendj,nh (found(p, q));

fi

Figure 3.15: Trail: Find Actions

presence of holes, we use a right hand rule, such as in [38], in order to route around

the hole and reach the destination. As indicated in the update algorithm for WSN

grid, during routing the node closest to, but outside a circle of radius 2k−1 around

ck−1 is marked as Nk−1. Since we assume that the network cannot be partitioned,

eventually such a node will be found. (If all nodes along the circle have failed, the

network is essentially partitioned).

Tolerating failures during a find: We now describe how the find message ex-

plores in squares of increasing levels. When a find message comes across a hole, it

is rerouted around the hole using geographic routing only radially outwards of the

current level square. If during the re-route, we reach a distance from the source of

the find corresponding to the next level of search, we continue the search at the next

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level and so on. Thus, in the presence of larger holes, we abandon the current level

and move to the next level, instead of routing around the hole back to the current

level of exploration.

(a) Find re-route along samelevel

(b) Find re-route to the nextlevel

Figure 3.16: Tolerating failures during find

Maintaining an existing trail: Nodes may fail after a trail has been created. In

order to stabilize from these states, we use periodic heartbeat actions along the trail.

We assume that if a node has a transient failure, the node returns in a null state.

We do not handle arbitrary state corruptions in the nodes.

Var

Hb : Interval for heartbeatsj.time : Local timej.sendHbp : Last time heartbeat was sentj.recvHbp : Last time heartbeat was receivedj.nextvertex : next high level vertex along Trail

Figure 3.17: Trail: Additional State at Mote j for Stabilizing Actions

91

The heartbeat actions are sent by each node along trailP to its child. At any

node r, if a heartbeat is not received from its parent, a search message is sent using

geographic routing tracing the boundary of the hole. trailP is reinforced starting from

the first node where the search message intersects trailP using a reinforce message

along the reverse path. (If the goal is to find trailP in the shortest time, the search

should likely be enforced in both directions along the boundary of the hole).

We have formally stated these maintenance actions in guarded command notation

in Fig. 3.18.

Tolerating failure of C: The terminating point C provides a sense of direction to

the trail and serves as a final landmark for the find operation. If C fails, the node

that is closest to C will takeover the role of C. However, even in the transient stage

when there is no C, the failure of C is tolerated locally. We describe this below.

Consider that C and all nodes in a contiguous region around C have failed. In

this case, a search message will be initiated from the node closest to C that belongs

to trailP . Because a contiguous set of nodes surrounding C have failed, the search

message eventually returns to the node initiating the search by following the bound-

ary of the hole. Thus an existing trailP terminates at the node that belongs to trailp

and is closest to C. Thus if a find message is unable to reach C, then routing along

the boundary of the hole will intersect trailP .

When a new node takes over the role of C, we do not assume that the state of the

original C is transferred. The new C has no knowledge of tracks passing through it.

Eventually, the maintenance actions for Trail will result in all tracks terminating at

C.

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Stabilizing Actions for Track Updates

〈S1〉 :: (j.childp 6= ⊥) ∧ (j.time − j.SendHbp = Hb) −→sendj,j.childp

(hearbeatp)reset j.SendHbp = j.time;

[]〈S2〉 :: recvk,j(heartbeatp) ∧ (j.prntp == k) −→

j.ReceiveHbp = j.time

[]〈S3〉 :: (j.prntp 6= ⊥) ∧ (j.time − j.ReceiveHbp = Hb) −→

sendj,nh (reroute(p, j, j.nextvertex))[]〈S4〉 :: recvk,j(reroute(p,m, n)) −→

if (j.childp 6= ⊥)send(j, nh) reinforce(p,m, j);

send(j, j.childp) clear(p);

j.childp = nh;else

sendj,nh (reroute(p,m, n))fi

[]〈S4〉 :: recvk,j(reinforce(p,m, n)) −→

if (j.childp 6= m)send(j, nh) reinforce(p,m, n);

j.childp = nh;j.prntp = k;

else

j.prntp = k;fi

Figure 3.18: Trail: Stabilizing Actions

3.3.4 Refinements of Trail

In Subsections 3.3.2 and 3.3.3, we have described the basic Trail protocol. In this

subsection, we discuss two techniques to refine the basic Trail network protocol: (1)

tuning how often to update a Trail tracking structure, and (2) tuning the shape of a

Trail tracking structure.

93

In the first refinement we alter the parameter b to values greater than 1. By

increasing b, we update the track of an object more often. This results in straighter

tracks with smaller stretch factor. As tracks get straighter, the find at higher levels of

the search can follow a triangular pattern (as illustrated in Fig. 3.21) as opposed to

complete circles. In fact, as b increases circular explorations can be avoided at more

levels of the find. Thus the average find cost in the network decreases as b increases.

In sum by increasing b, we increase the cost of update and decrease the cost of find.

This refinement can be used when the rate of updates is small as compared to the

rate of find. For example, as the speed of objects in the network decreases, we can

increase the value of b.

In the second refinement we change the length of each segment in Trail. In the

basic protocol, the segment at each level is a straight line to the next lower level. In

this refinement, we modify the length of each segment to be a straight line to the next

level plus an arc of length x × 2k. As x increases the amount of update at each level

increases, but the find exploration can now be smaller. Specifically when x = 2 × π,

the find is a straight line towards the center of the network. We call this particular

parameterization, the find-centric Trail protocol.

Tightness of Trail Tracking Structure

The frequency at which trailP is updated depends on parameter constant b in

property P3 of trailP . As seen in Section 3, for values of b > 1, trailP is updated

more and more frequently, hence leading to larger update costs. However, trailP

becomes tighter and tends to a straight line with the trail stretch factor approaching

1. We exploit this tightness of trailp to optimize the find strategy.

Optimization of find

94

We now describe the details of this optimization.

Lemma 3.3.9. Given trailP , (∠C, p, Nk) < (mx−k+1)∗(arcsin( 12b )), where (mx ≥

k ≥ 1).

Proof. Recall from Eq. 3.8, (∠p, Nj, cj) < (arcsin( 12b )), where Nj is any level j ver-

tex where mx ≥ j ≥ 1. Since Nj , Nj−1 and cj−1 form a straight line, recall that

∠NjpNj−1 + ∠pNjNj−1 = ∠pNjcj for mx > j ≥ 1. Similarly, since C, Nmx and cmx

form a straight line, also recall that ∠CpNmx + ∠pCNmx = ∠pNmxcmx. Using these

we have the following equations.

(∠NjpNj−1) < (arcsin(1

2b)) ∀j : (mx > j ≥ 1) (3.22)

(∠C, p, Nmx) < (arcsin(1

2b)) (3.23)

Using Eq. 3.22 and Eq. 3.23, we obtain ∠CpNk by summing up as follows.

(∠C, p, Nk) = ∠CpNmx +mx∑

j=k+1

(∠NjpNj−1)

< (mx − k + 1) ∗ (arcsin(1

2b))

From hereon, we let dpC denote the distance of any object P from C. After the

value of mx is defined for a trailP , object P can move for a certain distance before

mx is redefined. Therefore, given dpC, the value of mx in trailP cannot be uniquely

determined; however, we note that the value of mx can be bounded given dpC and

we define mxp as the highest possible value of mx in trailP , given dpC. We now

determine mxp.

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Let R denote the network radius, defined as the maximum distance from C. Recall

that mx denotes the number of levels in the track for an object P . mx is defined

as d(log(dist(C, po)))e − 1 where po is the position of the object when trailP was

(re)created from C. Given network radius R, let > be the highest number of levels

possible for any object in the network. Thus in a given network, > = d(log(R))e− 1.

Lemma 3.3.10. Given dpC, mxp = minimum(dlog(dpC)e,>).

Proof. Let mx be the index of the highest level in trailP . Using property P3 we get

that dist(p, cmx) < 2mx−b.

dist(C, cmx) ≤ dpC + dist(p, cmx)

< dpC + 2mx−b

By the definition of mx, 2mx < dist(C, cmx). Therefore we get the following

equation.

2mx < dpC + 2mx−b

Since b ≥ 1, we get the following equation.

dpC > 2mx−1

Thus mx ≤ d(log(dpC))e. Hence, mxp = minimum(dlog(dpC)e,>).

Since given dpC, the index of highest level mx in trailp cannot be uniquely deter-

mined, we state the maximum angle formed by ∠CpNk in terms of mxp rather than

96

mx. When the actual mx in trailP is lesser than mxp, then the actual maximum

angles formed by ∠C, p, Nk where 1 ≤ k ≤ mx is lower than the maximum angles

stated in the following equation.

(∠C, p, Nk) < (mxp − k + 1) ∗ (arcsin(1

2b)) (3.24)

In the analysis below, we characterize the minimum size of exploration required at

each level of exploration given the distance of finder object q from C. Only for ease

of explanation, we assume that b = 3.

Let Q be the finder at distance dqC from C. Thus mxq = minimum(dlog(dqC)e,>).

Let P be an object which should be found at the level k exploration. At level k of

the exploration, trailP for any location of P within the circle of radius 2k around q

should be intersected. A circular exploration of radius 2k around q is sufficient to

achieve this. We now characterize the necessary exploration.

We show that, at levels of exploration k where k ≥ mxq −7, circular explorations can

be avoided and instead a pattern of exploration along the base of an isosceles triangle

with apex q and length of base determined by Fig. 3.20 is sufficient to intersect the

trails of all objects at distance 2k from Q. The base of the isosceles triangle is such

that segment(C, q) is the perpendicular and equal bisector of the base of the triangle.

At levels of exploration k < mxq − 7, exploration along the entire circle is necessary.

Analysis of necessary exploration for optimized find algorithm Let Q be

the finder at distance dqC from C. Thus mxq = minimum(dlog(dqC)e,>). In the

basic find algorithm, the find operation will explore at all levels k where 0 ≤ k ≤

blog(dqC)c − 1 and at each level in circles of radius 2k. In terms of mxq, the highest

level of exploration for any location of q in the network is mxq − 1. This is because

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when mxq = >, it follows that mxq = blog(dqC)c and the highest level of exploration

in mxq − 1. When mxq = dlog(dqC)e, the highest level of exploration is mxq − 2.

Thus in either case, the level of exploration is bounded by mxq − 1.

Let P be an object which should be found at the level k exploration. At level k

of the exploration, trailP for any location of P within the circle of radius 2k around

q should be intersected. A circular exploration of radius 2k around q is sufficient to

achieve this. We now determine the necessary exploration.

Since dist(p, q) ≤ 2k , dpC ≤ dqC + 2k. Thus at any level k of the exploration,

mxp ≤ dlog(dqC + 2k)e. Note that dqC ≤ mxq and k ≤ (mxq − 1). Therefore

mxp ≤ mxq + 1.

We now outline our procedure for level of exploration k = mxq − 2.

Level of Exploration k = mxq − 2 Refer to Fig. 3.19. Let αb denote the value of

arcsin( 12b ) for a given b. Since in our analysis we consider b = 3, using Eq. 3.24 we

get that the maximum angle ∠CpNk is (mxp − k + 1) ∗ (α3). Since the finder object

Q is unaware of mxp, the worst case estimate for mxp is used, i.e. mxp = mxq + 1.

Thus, the maximum angle ∠CpNk = 3 ∗ α3.

Figure 3.19: Level of exploration k = mxq − 2

98

Given this angle, we are interested in determining the smallest segment(X”,X’)

that will intersect trailP for any location of P within the dotted circle. This is

obtained by drawing a segment from point p′ and p′′ at angle 3∗α3 with segment(C,p’)

and segment(C,p”) respectively. Point X ′ is obtained by extending this segment such

that dist(X ′, Y ′) = 2k. Point X ′′ is obtained similarly. Now segment(X’, X”) will

intersect trails of all objects at distance 2k from q, where k = mxq − 2.

From Fig. 3.19, we note that ∠qCp′ = arctan(14) and ∠Cp′X ′) = 3 ∗α3. Plugging

in the values we obtain the minimum required exploration at level k = ˆmxq − 2 as

follows (approximated to one decimal place):

dist(X ′, X ′′) = 2.5 ∗ 2k

Similarly, we determine the necessary pattern of exploration at levels 0, .., mxq −

1. Finally we show that, at levels of exploration k where k ≥ mxq − 7, circular

explorations can be avoided and instead a pattern of exploration along the base of an

isosceles triangle with apex q and length of base determined by Fig. 3.20 is sufficient

to intersect the trails of all objects at distance 2k from Q. The base of the isosceles

triangle is such that segment(C, q) is the perpendicular and equal bisector of the base

of the triangle. At levels of exploration k < mxq − 7, exploration along the entire

circle is necessary.

Note: All distances dqC in the range 2mxq−1 < dqC ≤ 2mxq , result in the same

value of mxq. But in the above analysis, we assumed dqC = 2mxq . This results

in finding the maximum exploration needed because when dqC is smaller, ∠Cp′X ′

increases, thus decreasing ∠V p′X ′ and lowering the length of exploration.

Optimized find algorithm (b=3):

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Exploration level k Length of triangle base Height of trianglemxq − 1 2 ∗ 2k 2k

mxq − 2 2.5 ∗ 2k 2k

mxq − 3 3.1 ∗ 2k 2k

mxq − 4 3.7 ∗ 2k 2k

mxq − 5 4.3 ∗ 2k 2k

mxq − 6 5 ∗ 2k 2k

mxq − 7 6.2 ∗ 2k 2k

Figure 3.20: Optimized find: pattern of exploration

1. Explore at levels k ranging from 0 to (blog(dqC)c−1). If k < (mxq −7), explore

using a circle of radius 2k around q. Else explore along the base of an isosceles

triangle with apex q and length of base determined by Fig. 3.20. The base of

the triangle is such that segment(C, q) is the perpendicular and equal bisector

of the base of the triangle.

An example for the modified find algorithm is shown in Fig. 3.21. In this figure, the

object q is at distance 48 units from C. mxq is 6. blog(dqC)c − 1 = 4. Therefore, the

levels of exploration are in the range 0..4. Exploration is along the base of triangles

at all levels. (This figure is not to scale but for illustration.)

Impact of the Optimization: The optimization of find at higher levels is thus

significant in that it yields: (1) smaller upper bounds for objects that are far away

from the finder; and (2) lower average cost of find(p, q) over all possible locations of

q and p.

As described earlier when b = 3, circular explorations are avoided at the highest 7

levels of the find operation. As the value of b increases, the number of levels at which

circular explorations can be avoided, increases. But by increasing b, we update the

100

Figure 3.21: Optimized find: example

track of an object more often. Thus by increasing b, we increase the cost of update

and decrease the cost of find.

We note that there are limits to tuning the frequency of updates, because for

extreme values of b distance sensitivity may be violated. For example, for large

values of b, that cause dist(p, ck) < y where y is a constant we end up with having to

update the entire trailP when an object moves only a constant distance y. Similarly,

for values of b < 0, the Trail Stretch Factor becomes unbounded with respect to

distance from an object. Thus an object could be only δ away from a point on trailP ,

yet the distance along trailP from this point to the p could travel across the network.

3.3.5 Modifying Trail Segments

The second refinement to Trail is by varying the shape of the tracking structure by

generalizing property P2 of trailP . Instead of trail segment k between vertex Nk and

Nk−1 being a straight line, we relax the requirement on trail segment k to be of length

at most (2 ∗ π + 1) ∗ 2k. By publishing information of P along more points, the find

path can be more straight towards C. An extreme case is when trail segment k is

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a full circle of radius 2k centered at ck and Seg(Nk, Nk−1). We call this variation of

Trail the find-centric Trail.

Figure 3.22: Find-centric Trail

Find-centric Trail

In this refinement, the find procedure eschews exploring the circles (thus travers-

ing only straight line segments) at the expense of the update procedure doing more

work. This alternative data structure is used when objects are static or when object

updates are less frequent than that of find queries in a system. Let trailP for object P

consist of segments connecting C, Nmx, .., N1, p as described before and, additionally,

let all points on the circles Circk of center ck and radius 2k contain pointers to their

respective centers, where mx ≥ k > 0.

Starting at q, the find path now is a straight line towards the center, as seen in

Fig. 3.22. If a circle with information about object P is intersected then, starting

from this point, a line is drawn towards the center of the circle. Upon intersecting

the immediate inner circle (if there is one), information about its respective center is

found, with which a line is drawn to this center. Object P is reached by following

102

this procedure recursively. We characterize the upper bound on the find cost in the

following Lemma.

Lemma 3.3.11. In find-centric Trail, when b = 1, the total cost of finding an object

P at point p from object Q at point q is 16 ∗ df where df = dist(p, q).

Proof. Let Q lie between circles of level k and k − 1 of the find-centric trail for P .

The worst case find cost occurs when q is just outside the level k−1 circle. Note that

dist(p, ck−1) < 2k−2 and therefore dist(p, q) > 2k−2

Now, q can travel distance 2 ∗ 2k to reach circle k. Let the point of intersection

of the find path from q and circle k be t. The cost of following pointers from t to the

centers of inner circles recursively and reaching P is given by (20 + 21 + ... + 2k), i.e.,

2 ∗ 2k.

The ratio of find cost to the distance is thus less than 4∗2k

2k−2 , i.e 16. Hence if

dist(p, q) = d, then the maximum cost of finding object P is 16 ∗ d.

We note that when events are static, the optimal publish structure is much smaller

than publishing along circular tracks. We have studied optimal publish structures for

querying in a static context in a related work [23].

Summary: In this section, we presented two refinements of Trail that lead to a

family of protocols. In the first refinement we altered the rate of updates. In the

second refinement, we altered the amount of updates at each level. One can choose

an appropriate parameterization depending on the expected rate of updates and finds

in the network.

103

As an example, given the expected rate of updates and expected rate of find op-

erations in the network, we can use the value of b in refinement 1 that minimizes the

sum of update costs and find cost over a given interval of time. We can compare this

cost with that of find-centric Trail and then choose the most appropriate parame-

terization. The find centric version of trail is especially beneficial when the rate of

updates is much smaller than that of finds.

3.3.6 Discussion

In our solution, we made some design decisions like choosing a single point to

terminate tracks from all points in the network and avoiding hierarchy in maintaining

the tracks. In this section, we analyze these aspects of our solution and compare them

with other possible approaches. We find that by avoiding hierarchy, we do not need to

partition the network into clusters and maintain these clusters, we can be more locally

fault-tolerant and we can obtain tighter tracks for any object. We also formally define

the notion of terminating points, differentiate those from clusterheads of a hierarchy,

and analyze the effect of more terminating points on the maximum find cost and

maximum update cost in the network.

Terminating points vs clusterheads: There are some hierarchy based solutions

[24, 27] for the problem of object tracking in a distance sensitive manner, where

the network is hierarchically partitioned into clusters and information of objects is

maintained at clusterheads1 at each level. Even in these solutions, information about

an object is published across the network to local clusterhead(s) at each level in the

hierarchy, all the way up to one or more clusterheads at the highest level in the

1Note that the responsibility of a clusterhead for every cluster could be shared by multiple nodesand not necessarily just one node. We refer to the abstraction of a leader for every cluster asclusterhead

104

hierarchy. We call these points at the highest level of the hierarchy as terminating

points.

Formally, a terminating set τ is a smallest set of points such that tracks of ob-

jects from every location in the network pass through at least one point in τ . The

cardinality of a set τ is denoted as µτ . There can be one or more terminating sets,

each with one or more terminating points. In Stalk [24] there is a unique clusterhead

at the highest level, thus there is a single terminating set with a single terminating

point. In LLS [1] and DSIB [27], there are multiple clusterheads at the highest level.

Thus there are multiple terminating sets, each with one terminating point. Tracks

from every point in the network pass through each of those clusterheads. Thus each

clusterhead at the highest level constitutes a terminating set by itself. In Trail there

is a unique terminating set with a unique terminating point, namely C.

It is in the process of maintaining tracks from a terminating point that we have

avoided hierarchy in Trail. In hierarchy based solutions, to maintain tracks and to

answer queries, tracks from terminating points necessarily pass through these cluster-

heads, where as Trail avoids hierarchy by determining anchors for the tracking paths

on-the-fly based on the motion of objects.

Merits of avoiding hierarchy: By avoiding hierarchical solutions we do not need

either a distributed clustering service that partitions the network into clusterheads

at different levels and maintains this clustering or a special (maybe centralized) allo-

cation of infrastructure nodes. By avoiding a hierarchy of such special nodes Trail is

also more locally fault tolerant. For example in the case of a find operation, failure

to retrieve information from an information server at a given level would require the

find to proceed to a server at the higher level [27]. This is particularly expensive at

105

higher levels of the hierarchy. On the other hand in Trail a find operation redirects

around a hole created by failed nodes using routing techniques such as the left hand

rule [38] and such faults can be handled, in a sense, proportional to the size of the

fault. Similarly, tracks to existing objects can be repaired more efficiently. As the

number of failures increase, there is only a proportional increase in find and update

costs.

Moreover, avoiding hierarchy allows for minimizing the length of tracking paths

given a terminating point. We analytically compare the performance of Trail with

that of other hierarchy based solutions for tracking objects in Section 3.5 and we

observe that Trail is more efficient than other solutions. Trail has about 7 times

lower update costs at almost equal find costs. By using a tighter tracking structure,

we are also able to decrease the upper bound find costs at larger distances and thereby

decrease the average find cost across the network.

Choice of terminating points: In Appendix B, we have formally analyzed the

choice of a unique terminating point for tracks from all points in the network and the

tradeoffs associated with multiple terminating sets and multiple terminating points

per set in terms of the maximum find and update costs in the network. We provide a

summary of our analysis here.

We show in our analysis that in order for find to be distance sensitive, it must

the case that all terminating points in a terminating set must be traversable in O(N)

where the network is of N × N dimensions. This precludes the possibility of track

from every point terminating at itself (or in other words, the terminating set being

equal to the set of all points in the network). So the question arises as to what are the

choices for number of terminating points and terminating sets. We consider 3 cases:

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a single terminating set with multiple terminating points, multiple terminating sets

each with one terminating point and a different terminating point for each type of

object.

Single terminating set with multiple terminating points: Intuitively, there exists a

possibility of decreasing the maximum track length in the network by dividing the

network into regions and having a local terminating point per region. The maximum

track length and therefore the maximum update cost in the network thus depends

on the size of the largest region. We are faced with the question of how small can

these regions be. We show that in order to maintain find distance sensitivity, the

diameter of the largest region can only be a constant order less than the diameter

of the network, i.e., at least Ω(N) in a N × N network. Thus, there can be only

a constant order of cost decrease compared to having only one terminating point.

Moreover, decreasing the maximum update cost by dividing the network into smaller

regions results in proportionate increase in the maximum find cost. This is because if

from any finder location, a track belonging to an object in any location in the network

is to be found, then it must be the case that the find trajectory contains all points in

the terminating set, thereby increasing the worst case find cost.

Multiple terminating sets each with one terminating point: If there are multiple ter-

minating sets then it is sufficient for find to traverse the terminating point in any

such set. In this case there is a likelihood of decreasing the maximum find cost when

compared to having only set of terminating points because tracks can be found by

reaching a terminating point in any of the terminating sets. The maximum find cost

in the network depends on the size of the largest region. However, we show in Ap-

pendix A that to maintain update distance sensitivity, the size of the largest region

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has to be at least Ω(N) in a N × N network. Also when the number of terminating

sets is greater than 1, update has to traverse the terminating point in all terminating

sets in the worst case. Thus the maximum update cost in the network increases.

Maintaining a track with respect to local terminating points could be advantageous

if it is more likely that querying object and the object being found are closer. Thus,

a find will never run into the scenario of having to traverse all regions in the network.

Similarly, maintaining a track with respect to multiple terminating point sets could be

advantageous if objects are likely to move within bounded regions within a network.

In this paper we consider all distances between querying object and tracked object

to be equally likely and do not restrict mobility of the objects. Hence we consider

only the case where there is a unique terminating set with a single terminating point,

namely C.

Different terminating point for each type of object: We note that it is also feasible

to select a different terminating point for different types of objects. In this paper we

describe how to maintain tracks for objects with respect to one terminating point

and guarantee find and update distance sensitivity. A different terminating point

can be chosen for each type of object based on hash functions and each type of finder

object can choose the respective terminating points as a worst case landmark; but this

concept is orthogonal to that of maintaining tracks with respect to a given terminating

point and is therefore compatible with Trail.

We now summarize some of the performance aspects of Trail in terms of load

balancing (fairness), maintenance and memory.

Load balancing and fairness: In hierarchy based solutions, to maintain tracks

and to answer queries, tracks from terminating points necessarily pass through these

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clusterheads, where as Trail avoids hierarchy by determining anchors for the tracking

paths on-the-fly based on the motion of objects. By avoiding hierarchy, in Trail load

is in fact more evenly distributed (more fairness) than hierarchical solutions in which

queries have to answered by specific locations and the same nodes are taxed.

Maintenance: The tracks maintained in Trail are almost straight with a stretch

factor of less then 1.2. By maintaining shorter tracks that are not convoluted, the

cost of maintaining the tracks in Trail is actually lower. We have also discussed in

detail the fault-tolerance actions of Trail in Section 4 where we describe how to handle

failures of nodes along existing tracks, failures during update and find and also how

failures of terminating point C can be handled locally.

Tolerating failure of the terminating point: In Section 3.3.3 we have shown

that in Trail, we can locally tolerate the failure of even C. Our protocol actions are

such that when C fails or a set of nodes in a contiguous region around C fail, track

for an object will terminate at any node that is closest to the boundary formed by

the hole. Thus during a find operation, a redirection rule as in GPSR is bound to

intersect the track and once again the faults are tolerated locally.

Memory efficiency: We note that the tracks maintained in Trail only contain

pointers to the current location of an object and not the state information of the

object. We also note that the storage required at each node is O(log(N)), per object,

where the network is of size N × N .

Handling bottleneck at C: We have previously discussed the option of having

multiple terminating sets, each with one terminating point. One example of this is

to construct a circle of constant radius δ around the center of the network and to

define each point on this circle as a terminating set. Thus tracks from any point in

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the network pass through all points on this circle. A find trajectory can intersect

any point on this circle to obtain a pointer to the object. The maximum update cost

increases by a factor of δ and the maximum find cost decreases by a factor of δ, and

still maintaining distance sensitivity. The responsibility of handling queries is now

distributed evenly around C.

3.3.7 Performance Evaluation

In this subsection, we evaluate the performance of Trail using simulations in JProwler

[70]. The goals of our simulation are: (1) to study the effect of routing stretch and

discretization errors on the trail stretch factor, (2) to study the effect of uniform node

failures on the performance of Trail, (3) to compare the average costs for find and

update, as opposed to the upper bounds we derived earlier, and (4) to analyze the

performance of Trail when scaled in the number of objects. Our simulation involves

a network of 8100 Mica2 motes arranged in a 90 × 90 grid.

Setup

Our simulation involves a 90 × 90 Mica2 mote network arranged on a grid. The

center of the network is placed at one corner, thus essentially simulating one quadrant

of the network. This setup lets us test the protocol over larger distances without

update and find operations reaching the center. We use JProwler as our simulation

platform with a Gaussian radio fading model. The mean interference range is a grid

area of 5 × 5 square units. Packet transmission time is set at 40 ms. We implement

geographic routing on a grid to route messages in the network. In the presence of

failures we use a left hand rule to route around the failure [38]. We assume an

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underlying link maintenance layer because of which the list of up neighbors is known

at each node.

Performance of update operations

We determine the number of messages exchanged for object updates over differ-

ent distances when an object moves continuously in the network. We consider the

unit grid separation, where each node has at most 4 communication neighbors. The

number of neighbors may be lesser due to failures. We calculate the amortized cost

by moving an object in different directions and then observing the cumulative num-

ber of messages exchanged up to each distance from the original position to update

the tracking structure. The results are shown in Fig. 3.23(a). The jumps visible at

distances 4 and 8 show the impact of the log(d) factor in the amortized cost. At

these distances, the updates have to be propagated to a higher level. We also study

the effect of uniform failures in the network on the increase in update costs. We

consider fault percentages up to 20. We see from the figure that even with failures

the average communication cost increases log linearly with distance. This indicates

that the failures are handled locally.

Trail stretch factor

From Section 3.3.2, we note that in the continuous model, for an object P at

distance dpC from C, length of trailP is less than 1.2 ∗ dpC. We now study the

effect of routing overhead and the discretization factor on the length of the tracking

structure that is created. We measure the trail length in terms of the number of hops

along the structure. Fig. 3.23(b) shows the average ratio of distance from C to the

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(a) Trail update cost (amortized) (b) Trail stretch factor

Figure 3.23: Trail update costs and Trail stretch factor

length of the trail during updates over different distances from the original position.

The parameter b = 1 in these simulations.

When the trail is first created, the trail stretch factor is equal to the routing stretch

factor from C to the original location. In the absence of failures, we notice that the

trail stretch factor is around 1.4 at updates of smaller distances and then starts

decreasing. This can be explained by the fact the trail for an object starts bending

more uniformly when the update is over a large distance. Even in the presence of

failures, the trail stretch factor increases to only about 1.6 times the actual distance.

Performance of find

We first compare the average find costs of Trail with upper bounds derived. We

study this in the presence of no interference, i.e. there is only one finder in the

network. We fix the finder at distance 40 units from C. We vary the distance of

object being found from 2 to 16. We evaluate using the basic find algorithm with

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b = 1. In Fig. 3.24(a) and Fig. 3.24(b), we show the average number of messages and

the average latency for the find operation respectively.

(a) Average messages (find) (b) Average latency (find)

Figure 3.24: Trail: find cost

The analytical upper bound 38 ∗ d (obtained from Lemma 3.3.8 for b = 1) is

indicated using dotted lines in Fig. 3.24(a), and we see that the number of messages

exchanged during find operations are significantly lower. When there is only one finder

in the network, there is no interference. Therefore there are no re-transmissions except

for the case when there is a loss due to probabilistic fading. Therefore the latencies

are roughly equal to the number of messages times the message transmission time per

hop. The jumps at distances 3, 5 and 9 are due to increase in levels of exploration

at these distances. The results of the above simulations thus validate our theoretical

bounds derived in Section 3 and 4.

Impact of interference

We now evaluate the effect of interference when multiple objects are present in

the network. Note that Trail operates in a model where queries are generated in

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an asynchronous manner. We first evaluate the effect on find latency when objects

are uniformly distributed across the network. We then evaluate the performance in

a more severe environment where all the objects being found are collocated in the

network.

In the presence of interference, messages are likely to be lost and we have im-

plemented the following reliability mechanism to counter that. Forwarding a find

message by the next hop is used as an implicit acknowledgment. The find mes-

sages are retransmitted up to 4 times by each node. The interval to wait for an

acknowledgment is doubled after every retransmission starting with 100 ms for the

first retransmission. Note that 100 ms is a little more than twice the transmission

time for each message. Also upon sensing traffic, the MAC layer randomly backs off

the transmission within a window of 10 ms. The maximum number of MAC retries

are set to 3.

(a) Average latency (find) (b) Loss percentage (find)

Figure 3.25: Effect of interference on find cost

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In the first scenario we uniformly distribute the objects in a 50 × 50 area in the

center of the quadrant being simulated. By distributing the objects in the middle of

the quadrant being simulated, we avoid the decrease in find messages simply because

an object is close to the boundary. We simulate 2, 10, 20, 30, 40 and 50 objects

in the network. We observe no significant increase in latency from 2 objects up to

30 objects. For the case of 40 and 50 objects in the network we observe increase

in average latency especially at larger distances. At larger distances, find messages

from different objects interfere to a significant extent. Despite messages being re-

transmitted up to 4 times, we also see losses during the find operation at distances

greater than 12 units. The latency and loss percentages at different distances for 2,

40 and 50 objects are shown in Fig. 3.25(a) and Fig. 3.25(b) respectively.

(a) Average latency (find) (b) Loss percentage (find)

Figure 3.26: Effect of interference on find cost: objects being found collocated

We now consider a more severe scenario where all the objects being found are

at the same location. We compute the average latency for the find operation when

objects issuing find query are uniformly distributed around this location, at different

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distances. Fig. 3.26(a) shows the latency with respect to distances for 2, 14 and 18

objects when all objects being found are at the same location. Fig. 3.26(b) shows

the loss percentages for 2, 10, 14 and 18 objects when all objects being found are

at the same location. As expected, we see from Fig. 3.26(a) and Fig. 3.26(b) that

interference is severe at smaller distances. We see loss percentages as high as 60%

when there are 18 objects at small distances.

Summary of evaluation

We observe from the above figures that Trail has a find time that grows linearly

with distance. When scaled in the number of objects up to 50, with objects uniformly

distributed in a 50×50 area and concurrently issuing queries, the query response time

still does not increase substantially. However at a scale of 40 objects and distances

of greater than 12 units we observe losses of around 10% during the find operation.

This is because, at larger distances find messages from different objects interfere to a

significant extent. In a potentially more severe scenario where all objects being found

are at the same location and objects issuing find are distributed uniformly around

that location, interference is significant at smaller distances. We see loss percentages

as high as 60% when there are 9 pairs of objects at small distances.

3.4 Implementation of Trail in a Real Network

We have implemented Trail for the special case of a long linear topology network

for demonstrating an intruder interceptor tracking application. We have experimen-

tally validated the performance of this version of Trail on 105 Mica2 motes in the

Kansei testbed [3] under different scaling factors such as the number of objects in the

system, the frequency of queries, and the speed of the objects in the network. This

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implementation has been used to support a distributed intruder interceptor tracking

application where the goal of the interceptor is to catch the intruders as far away

from an asset as possible. This application was demonstrated at Richmond Field

Station in Berkeley in 2005 as part of the DARPA NEST Program. In this section,

we describe the results of these experiments.

3.4.1 Experimental setup

We use a network of 105 XSM-Stargate pairs in a 15 × 7 grid topology with 3

ft spacing in the Kansei testbed. The XSMs are a Mica2 family of nodes with the

same Chipcon radio but with additional sensors mounted on the sensor board. The

XSMs are connected to Stargate via serial port and the Stargates are connected via

Ethernet in a star topology to a central PC. We are able to adjust the communication

range by adjusting the power level and the XSMs can communicate reliably up to 6

ft at the lowest power level but the interference range could be higher. Trail operates

asynchronously with no scheduling to prevent collisions. Hence, we implement an

implicit acknowledgment mechanism at the communication layer for per-hop relia-

bility. The forwarding of a message acts as acknowledgment for the sender. If an

acknowledgment is not received, then messages are retransmitted up to 3 times.

Object traces

We now describe how the object motion traces are obtained. Motes were deployed

in a grid topology with 10 m spacing at Richmond Field Station. Sensor traces were

collected for objects moving through this network at different orientations. Based

on these traces, tracks for the objects are formed using a technique described in [2].

These tracks are of the form (timestamp, location) on a 140m × 60m network. These

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object tracks are then converted to tuples of the form (id, timestamp, location, grid

position) where grid position is the node closest to the actual location on the 15 × 7

network and id is a unique identifier for each object. These detections are injected

into the XSM in the testbed corresponding to the grid position via the Stargate at

the appropriate time, using the injector framework in Kansei. Thus, using real object

traces collected from the field and using the injector framework, we emulate the object

detection and association layer to evaluate the performance of our network service.

3.4.2 Results

We evaluate the performance of Trail under different scaling factors such as in-

creasing number of objects, higher speed of objects and higher query frequency in

terms of the reliability and latency of the service. We run Trail with 2, 4, 6 and 10

mobile objects, in pairs. One object in each pair is the object issuing find query and

the other object is the object being located. In each of these scenarios, we consider

query frequency of 1 Hz, 0.5 Hz, 0.33 Hz and 0.25 Hz. The object speed affects the

operation of Trail in terms of the rate at which grow and clear messages are gener-

ated. We consider 3 different object update rates, one in which objects generate an

update every 1 second, every 2 seconds and every 3 seconds. Considering that the

object traces were collected with humans walking across the network acting as objects

with average speed of about 1 m/s, object update rates of 1 Hz and 0.5 Hz enable a

tracking accuracy of 1m and 2m respectively.

In the 4, 6 and 10 objects scenario, we consider a likely worst case distribution

of the objects where all objects issuing find and all objects being found are within

communication range. Moreover, as optimal pursuit control requirements suggest

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[15], the query frequencies depend on relative locations and are lesser when objects

are far apart, but we consider all objects issuing queries at the same frequency. If

the replies are not received before the query period elapsed, then the message is

considered lost. The loss percentages are based on 100 find queries at every distance

and the latencies are averaged over that many readings.

Scaling in number of objects

Fig. 3.27 shows the latency and loss for find operations as the number of objects

increases with query frequency fixed at 0.33 Hz and object updates fixed at 0.5 Hz.

Fig. 3.28 shows the latency and loss for find operations as the number of objects

increases with query frequency fixed at 0.5 Hz and object updates fixed at 0.5 Hz.

Scaling in query frequency

Here we analyze how the latency and reliability of Trail are affected as the query

frequency increases. In Fig. 3.29, we show the reliability and latency of Trail with 6

objects under query frequencies of 1 Hz, 0.5 Hz, 0.33 Hz and 0.25 Hz, with object

update rate of 0.5 Hz.

Scaling in object speed

Fig. 3.30 shows the latency and loss for find operations with increasing object

speeds that generate updates at 0.33 Hz, 0.5 Hz and 1 Hz. The query frequency is

0.5 Hz and the number of objects is 6.

Summary of experimental evaluation

We observe from the above figures that Trail offers a query response time that

grows linearly with the distance from an object. Trail operates in an environment

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(a) Latency of Trail (b) Loss ratio

Figure 3.27: Scaling in number of objects (query frequency 0.33 Hz, object update 0.5 Hz)

(a) Latency of Trail (b) Loss ratio

Figure 3.28: Scaling in number of objects (query frequency 0.5 Hz, object update 0.5 Hz)

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(a) Latency of Trail (b) Loss ratio

Figure 3.29: Scaling in query frequency (6 objects, object update rate 0.5 Hz)

where objects can generate updates and queries asynchronously and in such an envi-

ronment, interference increases the response time. From our experiments, we observe

that query latency and loss percentages increase with number of objects and speed

of objects but the loss ratio is not severe. As seen in Fig. 3.27, scaling the number

of objects up to 10 yields a loss rate of up to 7% with a query frequency 0.5 Hz and

an object update rate of 0.5 Hz. As seen in Fig. 3.30, scaling the object speeds to

generating 1 update per second results in a loss rate of up to 7 % even with 6 objects

in the system and query frequency of 0.5 Hz. Increasing query frequencies has a more

severe impact on loss percentages especially with more objects in the network. In

Fig. 3.29, we notice that with 6 objects in the network, loss increases substantially as

the query frequency becomes 1 Hz; this happens due to higher interference leading to

congestion.

Handling interference: To tolerate network interference, spatial and temporal

correlations that exist in the application can be exploited in the following way. The

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rate at which information is needed by the pursuer is known. The network service

can be notified of the next instant at which the state of the evader is needed and the

location where the query results need to be sent. The query results can then arrive

just in time. Advance knowledge about the query and the deadline can be used to

decrease the interference in the network when multiple pursuers query about evaders

in the network and more so when the objects are densely located. Another possibility

to deal with interference is a synchronous push version of the network tracking service

where snapshots of objects are published to subscribers in a distance sensitive manner

thereby avoiding interference. By the same token, applications should be aware of

other extreme conditions (in terms of object number and speed) for effectively using

the service. For example, applications may compensate for losses by increasing their

query frequency, but this should account for extreme scenarios where the increased

frequency itself results in higher interference.

(a) Latency of Trail (b) Loss ratio

Figure 3.30: Scaling in object speed (6 objects, query frequency 0.5 Hz)

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3.5 Related Work

In this section, we discuss related work and also compare the performance of Trail

with other protocols designed for distance sensitive tracking and querying.

Tracking: As mentioned earlier, mobile object tracking has received significant

attention [5, 24, 30] and we have focused our attention on WSN support for tracking.

Some network tracking services [25] have non-local updates, where update cost to

a tracking structure may depend on the network size rather than distance moved.

There are also solutions such as [5, 24, 1] that provide distance sensitive updates and

location.

Locality Aware Location Services (LLS) [1] is a distance sensitive location service

designed for mobile ad-hoc networks. In LLS, the network is partitioned into hierar-

chies and object information is published in a spiral structure at well known locations

around the object, thus resulting in larger update costs whenever an object moves.

The upper bound on the update cost in LLS is 128 ∗ dm ∗ log(dm), where dm is the

distance an object moves, as opposed to the 14 ∗ dm ∗ log(dm) cost in Trail; the upper

bounds on the find cost are almost equal. Moreover, as seen in Section 5, we can

further reduce the upper bound on the find cost at higher levels in Trail.

The Stalk protocol [24] uses hierarchical partitioning of the network to track ob-

jects in a distance sensitive manner. The hierarchical partitioning can be created with

different dilation factors (r ≥ 3). For r = 3 and 8 neighbors at each level, at almost

equal find costs, Stalk has an upper bound update cost of 96∗d∗ log(d). This increase

occurs because of having to query neighbors at increasing levels of the partition in

order to establish lateral links for distance sensitivity [24].

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Both Stalk and LLS use a partitioning of the network into hierarchical clusters

which can be complex to implement in a WSN, whereas Trail is cluster-free. Moreover,

in Stalk, the length of the tracking structure can span the entire network as the

object keeps moving and, in LLS, the information about each object is published in a

spiral structure across the network. In comparison, Trail maintains a tighter tracking

structure (i.e., with more direct paths to the center) and is thus more efficient and

locally fault-tolerant.

Figure 3.31: Trail: analytical comparison

In [5], a hierarchy of regional directories is constructed and the communication

cost of a find for an object df away is O(df ∗ log(N)) and that of a move of distance

dm is O(dm ∗ log(D) ∗ log(N)) (where N is the number of nodes and D is the network

diameter). A topology change, such as a node failure, however, necessitates a global

reset of the system since the regional directories depend on a non-local clustering

program that constructs sparse covers.

Querying and storage: Querying for events of interest in WSNs has also received

significant attention [36, 63, 52] and some of them focus on distance sensitive querying.

We note that Trail, specifically the find-centric approach can also be used in the

context of static events.

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In [52], a balanced push-pull strategy is proposed that depends on the query

frequency and event frequency; given a required query cost, the advertise operation is

tuned to do as much work as required to satisfy the querying cost. In contrast, Trail

assumes that query rates depend on each subscriber (and potentially on the relative

locations of the publisher and subscriber), and it also provides distance sensitivity

during find and move operations, which is not a goal of [52]. In directed diffusion

[36], a tree of paths is created from all objects of interest to the tracker. All these

paths are updated when any of the objects move. Also, a controller initiated change in

assignment would require changing the paths. By way of contrast, in Trail, we impose

a fixed tracking structure, and tracks to all objects are rooted at one point. Thus,

updates to the structure are local. Rumor routing [14] is a probabilistic algorithm to

provide query times proportional to distance; the goal of this work is not to prove a

deterministic upper bound. Moreover, its algorithm does not describe how to update

existing tracks locally and yet retain distance sensitive query time when objects move.

Geographic Hash tables [63] is a lightweight solution for the in-network-querying

problem of static events. The basic GHT is not distance sensitive since it can hash

the event information to a broker that is far away from a subscriber. The distance

sensitivity problem of GHT can be alleviated to an extent by using geographically

bounded hash functions at increasing levels of a hierarchical partitioning as used in

DIFS protocol. Still, attempting such a solution suffers from a multi-level partitioning

problem: a query event pair nearby in the network might be arbitrarily far away in

the hierarchy. However, we do note that GHT provides load balancing across the

network, especially when the types of events are known and this is not the goal of

Trail.

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Distance Sensitive Information Brokerage [27] protocol performs a hierarchical

clustering of the network and information about an event is published to neighboring

clusters at each level. Using Find-centric Trail we can query information about static

events in a distance sensitive manner. We also note that when events are static, the

optimal publish structure is much smaller than publishing along circular tracks. We

have studied optimal publish structures for querying in a static context in a related

work [23].

Spatio-temporal query services Motivated by a class of applications in which

mobile users are interested in continuously gathering information in real time from

their vicinities, a network data service called spatio temporal query has been proposed

in [53]. The spatial constraint for the network service comes from an energy efficiency

point of view; only nodes relevant to a query area should be involved in contributing

the query result. The temporal constraint is due to a requirement on data freshness

for the query results. An approximate motion model for the mobile user is assumed.

Specifically the motion can be predicted over a small interval of time. The query area

at any time is a function of the current location of the mobile user. The key difference

in Trail is that the query area in consideration is the entire network as opposed to a

function of the querier location as in [53].

We note that, when Trail is used in the context of a distributed pursuer evader

game, spatial and temporal correlations that exist in the application can be exploited

using ideas presented in [53] to improve the performance of the application and the

network. The rate at which information is needed by the pursuer is known. The

network service can be notified of the next instant at which the state of the evader is

needed and the location where the query results need to be sent. The query results

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can then arrive just in time. Constraints on evader speed can be exploited as follows.

Subsequent queries for evader location can originate from the previous evader location

and the results can be routed back to the pursuer.

3.6 Summary

In this chapter, we considered a pursuer-evader game known in the literature

as ”asset protection”, and formulated optimal strategies under communication con-

straints, established bounds on the information requirements of these strategies, and

derived scaling laws for these bounds. In particular, we showed that the min-max

optimal pursuer strategy of the full information game extends to networked games,

provided that the sampling period and the delay for the evader state information up-

dates scale linearly with the pursuer-evader distance. We proposed a novel min-max

equilibrium concept for networked differential games by introducing an omniscient

opponent which can maximally exploit the delays and the inter-sample periods in the

information state updates.

We then presented Trail, a family of protocols for distance sensitive distributed

object tracking in WSNs. Trail avoids the need for hierarchical partitioning by de-

termining anchors for the tracking paths on-the-fly, and is more efficient than other

hierarchy based solutions for tracking objects: it allows 7 times lower updates costs

at almost equal find costs and can tolerate faults more locally as well.

Importantly, Trail maintains tracks from object locations to only one terminating

point, the center of the network. Moreover, since its tracks are almost straight to the

center with a stretch factor close to 1, Trail tends to achieve the lower bound on the

total track length. By using a tight tracking structure, Trail is also able to decrease

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the upper bound find costs at larger distances and thereby decrease the average find

cost across the network.

Trail is a family of protocols and we have shown that refinements of the basic

Trail protocol are well suited for different network sizes and query frequency settings.

We have validated the distance sensitivity and fault tolerance properties of Trail in a

simulation of 90×90 network using JProwler. We have also successfully implemented

and tested the Trail protocol in the context of a pursuer evader application for a

medium size (over 100 node) mote network.

Trail operates in an environment where objects can generate updates and queries

asynchronously. We note that in such an environment, due to the occurrence of

collisions, there can be an increase in the message complexity for querying and updates

especially when the objects are densely located in the network. As future work, we

are considering a push version of the network tracking service where snapshots of

objects are published to subscribers in a distance sensitive manner, both in time and

information, in order to increase the reliability and energy efficiency of the service

when the density of objects in the network is high.

We also note that spatial and temporal correlations that exist in the application

can be exploited to improve the performance of the application and the network.

Two such examples are as follows. (1) The rate at which information is needed by

the pursuer is known. The network service can be notified of the next instant at which

the state of the evader is needed and the location where the query results need to be

sent. The query results can then arrive just in time. Advance knowledge about the

query and the deadline can be used to decrease the interference in the network when

multiple pursuers query about evaders in the network and more so when the objects

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are densely located. (2) Constraints on evader speed can be exploited as follows.

Subsequent queries for evader location can originate from the previous evader location

and the results can be routed back to the pursuer. Thus the energy efficiency of the

network can be improved.

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CHAPTER 4

CLASSIFICATION OF INTRUDERS AND TRACKMONITORING

In this chapter, we study the application of sensor networks to the problems of

classifying and monitoring tracks of targets. We consider a surveillance application

scenario, whose objective is to identify a breach along a perimeter or within a re-

gion. The intruding object, or target, may be an unarmed person, a soldier carrying

a ferrous weapon, or a vehicle. The three fundamental user requirements of this

application are target detection, classification, and tracking.

Detection requires that the system discriminate between a target’s absence and

presence. Successful detection requires a node to correctly estimate a target’s presence

while avoiding false detections in which no targets are present. The key performance

metrics for detection include the probability of correct detection, and the probability

of false alarm.

Classification requires that the target type be identified as belonging to one of

several classes including person, soldier, and vehicle. More generally, classification is

the result of M-ary hypothesis testing and depends on estimation, which is the process

of determining relevant parameters of the detected signal including, for example, its

peak amplitude, phase, duration, power spectral density, etc. Successful classification

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requires that targets are labeled by the system as being members of the class to which

they actually belong.

Tracking involves maintaining the target’s position as it evolves over time due

to its motion in a region covered by the sensor network’s field of view. Successful

tracking requires that the system estimate a target’s initial point of entry and current

position with modest accuracy and within the allowable detection latency. Implicit

in this requirement is the need for target localization. The tracking performance

requirements dictate that tracking accuracy, or the maximum difference between a

target’s actual and estimated position, be both bounded and specified, within limits,

by the user. The system is not required to predict the target’s future position based

on its past or present position.

We design a dense, distributed, and 2-dimensional sensor network-based classifi-

cation and tracking system using inexpensive sensor nodes. In this model, intrusion

data are processed locally at each node, shared with neighboring nodes if an anomaly

is detected, and communicated to an ex-filtration gateway with wide area networking

capability. The motivation for this approach comes from the spatial- and temporal-

locality of environmental perturbations during intrusions, suggesting a distributed

approach that allows individual sensor nodes, or clusters of nodes, to perform local-

ized processing, filtering, and triggering functions. Collaborative signal processing

enables the system to simultaneously achieve better sensitivity and noise rejection,

by averaging across time and space, than is possible with an individual node which

averages only across time.

Our approach thus demonstrates how dense, resource-constrained sensor networks

yield improved spatial fidelity of sampling the environment. More specifically, we

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introduce a spatial statistic called the influence field, realize an estimator for it using

a binary sensor field, and use it as the basis for a new type of classifier and tracker.

The influence field of an object thus depends on the type(s) of sensors being used for

detection (magnetometers in Fig. 4.1), and it is essentially characterized by the area

and the shape of the region. Figure 4.1 illustrates the differences between magnetic

influence fields for two objects, a person carrying a metal rod and a vehicle. The

size and shape of the influence fields shown in this figure depend on the amount and

distribution of metallic content in each object type and the orientation of the object

(e.g., the dumbbell shape of the vehicle influence field is attributed to the positions

of its axles). Each sensor node merely has to detect a binary “presence” of an object;

network-based aggregation of these bits yields the influence field without requiring

substantial or complex node operation.

Figure 4.1: Magnetometer based influence fields for two object types.

The key challenge in realizing the influence field is the unreliability of wireless

sensor networks. Event loss –both in nodes and in the network– is fundamental to

wireless sensor node platforms and its impact on the application can be substantial.

Thus both node and network unreliability have to be dealt with in estimating the

influence field.

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In Section 4.1, we state our system model, describe our approach for classification

and tracking by estimation of influence fields. In Section 4.2, we derive necessary

conditions for reliably estimating the area and shape of various objects’ influence

field in a manner that preserves their difference despite faulty detection, network level

fading and channel contention. We then design network parameters and algorithms to

meet these conditions. In Section 4.3, we describe how our results and techniques can

be composed and applied to identify and track objects. We have demonstrated our

influence field based classification and tracking approach in the context of 2 sensor

network based surveillance applications namely, A Line in the Sand (18m by 7m area

and 100 nodes), and ExScal (1Km by 200m area and over 1000 nodes). In Section 4.4,

we provide some details of the implementation of our techniques in the context of a

surveillance application demonstration, A Line in the Sand, and show how we dealt

with different fault classes in our design. In Section 4.5, we describe extensions to our

approach using multiple sensing modalities. In Sec. 4.6, we describe previous work

related to ours. In Section 4.7, we present a summary.

4.1 Influence Field Based Classification and Tracking

In this section, we state the system and fault model and describe our solution

for classification and tracking by estimation of influence fields. We then identify the

challenges in reliably estimating the influence field.

4.1.1 System Model

The system consists of N wireless sensor nodes, each with a unique identifier. We

assume a localization service that provides the relative or absolute position for each

node and a global time synchronization service that enables each node to timestamp

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its detections. The sensor nodes are distributed uniformly over a geographic region

that is to be monitored. We model this region as a large finite number, Ω, of per-

fectly spaced logical points that serve as the potential locations where nodes can be

deployed. When we refer to the area A of a subregion, we mean A is the number of

these Ω points in the subregion. We denote the ratio N/Ω by ρ, which represents the

sensor density in the network.

We assume that the wireless network is connected, hence it is possible to aggregate

messages from any subset of the nodes in the system, albeit that such aggregation may

require multi-hop communications. If this is the case, we assume a central aggregator

node that is known to the rest of the nodes.

Recall that the influence field of an object j with respect to a given sensing

modality is the region surrounding j where j will be “detected” by a sensor of that

type located at any point in that region. For simplicity of presentation, we assume

that the size of the influence field of j is invariant with respect to its location. Likewise,

the shape of the influence field is invariant with respect to location, up to rotation.

We limit our attention to one given sensing modality for simplicity. The modality

will remain implicit in our notation.

4.1.2 Fault Model

Sensor networks are subject to a large class of faults, resulting from inexpensive

hardware, limited resources, unreliable communications and extreme environmental

conditions. We consider both node and network fault types.

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Node faults

Sensor nodes fail in a variety of ways, including hardware and software failures,

or simply in the form of a transient event loss. They also occasionally generate false

positives due to unreliable hardware, environmental perturbations and transient state

corruption. Thus at any time, the net effect of a node fault can be modeled as missing

the detection of an object, i.e., a false negative, or asserting a detection when there

is no object, i.e., a false positive.

Node fault model: The probability that any node misses the detectionof an object, whether it is due to a transient, permanent or intermittent

node fault, is 1−pn, while the probability that any node generates a falsepositive is pfp.

In other words, we assume that false negatives and false positives at nodes are

independent of each other and of the objects. It follows trivially that for any mean-

ingful detection to be possible, the probability of a node detecting an event, pn, must

be greater than the probability of a node generating a false positive, pfp.

Network faults

Wireless communications in a multi-hop network are subject to both fading and

contention effects. Channel fading loss depends on link characteristics such as dis-

tance, relative orientation of sender and receiver, environmental conditions, etc.

Fading model: The probability of message loss due to fading for anysingle hop communication in the network is 1 − pf . It follows that in the

absence of any other faults, the probability of message loss due to fadingfor any h hop communication is 1 − ph

f .

Channel contention losses occur when multiple senders try to transmit a message

at the same time. The degree of message loss depends on several factors like the

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type of Medium Access Control (MAC) protocol used, the inherent synchronicity of

message transmissions, and most importantly the number of nodes trying to send data

simultaneously. We assume a standard CSMA/CA MAC protocol wherein nodes try

to avoid collisions using random backoffs and channel sensing.

Contention model: The probability of message contention loss, which is

a function of the number of nodes simultaneously trying to send a messageand of the number of slots available for a node to choose its random backoff

from, is 1 − pc.

The end-to-end network reliability of message reception for a given source depends

on several factors including the size of the traffic load, the number of hops traversed

by each message and the routing protocol. We assume a convergecast routing model,

wherein each message is forwarded by a node to its next hop neighbor until it reaches

the aggregator.

End-to-end reliability model: The probability of end-to-end message

loss for a traffic source i, which is a function of the number of senders ini, and the distance or number of hops h between the traffic source and

the aggregator, is 1 − prcvih

.

Our model postulates that network unreliability is uniform across nodes equidis-

tant from the aggregator. It is possible to realize the aforementioned end-to-end

reliability model by careful design of the routing protocol used. The GridRouting

protocol, achieves this by uniformly balancing the trac load over paths including

stable, reliable links.

4.1.3 Estimating the influence field

Estimation consists of calculating the area and the shape of the influence field.

Recall that the distribution of sensor nodes is uniform. If we assume sensor node den-

sity ρ exceeds some lower bound that depends on the targets at hand, the estimation

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of the area A of the influence field of j is effectively reduced to counting the number of

nodes that detect j. With uniform distribution, the number of sensors in any region

of area A follows a binomial distribution with parameters (A, ρ). For practical values

of A and ρ, we can exploit the rule of thumb for the normal approximation to this

binomial distribution that the value of the random variable lies within 3 times the

standard deviation of the expected value in 99% of the trials.

Proposition 1 Given uniform deployment of sensors, with high probability (whp),

the number of sensors that lie in the influence field of j is in the interval

(A × ρ) − 3 ×√

A × ρ × (1 − ρ) .. (A × ρ) + 3 ×√

A × ρ × (1 − ρ)

Relation to sensor coverage. Sensor coverage in a region is the minimum number of

sensors that “cover” (i.e., will detect) each point in the region. While this typical

definition of the concept is independent of the type of objects at hand, a more useful

definition for our purposes would be one that is with respect to each object type.

(Thus, the sensor coverage of object j in a region may be different from the sensor

coverage of another object in that region.)

Classification is an example of an application that can exploit area estimation.

Different object types may be classified via separation between the areas of their re-

spective influence fields. Errors in area estimates can thus result in mis-classifications.

Tracking is an example of an application that can exploit shape estimation. Object

location may be tracked from the locations of the sensors that detect it, i.e., by

computing the centroid of their locations. Shape distortion errors can thus result in

inaccuracy of tracking. (The same argument would apply for classification based on

the shape of the influence field.)

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Node and network faults impact estimation of the area and the shape of the

influence field of j. In particular, the impact may not be simply proportional to the

area and the distribution of the sensors whose detections are successfully aggregated

may no longer be uniform.

We are therefore led to the problem of how to reliably estimate the size and shape

of the influence fields of objects so that they can still be compared. More specifically,

we focus on two subproblems of reliable estimation:

1. How to ensure that for objects whose respective influence field areas are sepa-

rable, the separation remains between the fault-affected estimates of their re-

spective influence field areas?

2. How to preserve the shape of the influence field of an object in the fault-affected

estimate of the object influence field, by preserving the uniformity of distribution

of the sensors whose detections are not affected?

4.2 Reliable Estimation Despite Faults

In this section, we address problems 1 and 2 outlined above successively for each

of the fault models discussed. Specifically, for each fault model,

• we analytically identify necessary conditions for accurately preserving the sep-

aration between area estimates and the shape estimates,

• we derive an ideal density for node deployment with respect to the given sensing

modality for the efficiency of estimation, and

138

• we characterize algorithmic techniques and parameter tuning options to deal

with situations where the ideal density is not achievable, including cases where

the deployed density is less than and more than ideal.

Also, where appropriate, we provide experimental corroboration of our analysis

and algorithmic techniques via experiments realized on Mica2 sensor motes. Our

approach is compositional and thus the analysis, which is presented separately for

each fault type, can build upon the constraints/distributions identified for other fault

types.

4.2.1 Tolerating false reports

We model the net effect of node faults is modeled as false negatives and false

positives.

False Negatives

Let A1, A2, ... Ak be the influence fields of k types of objects ranging from the

smallest to the largest. Recall that pn is the probability that a sensor node detects

an object. The number of non-faulty nodes in a region of area Ai thus has a binomial

distribution with parameters (ni, pn), where ni is the number of nodes in the area

Ai. However, recall from Proposition 1 that ni itself is a random variable that has a

binomial distribution, characterized by the following equation.

E(ni) = ρ × Ai (4.1)

V (ni) = Ai × ρ × (1 − ρ) (4.2)

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The mean and variance of the number of nodes detecting object i, E(di) and V (di)

respectively are given by the following equations.

E(di) = ρ × Ai × pn (4.3)

V (di) = Ai × ρ × pn × (1−ρ × pn) (4.4)

For this distribution, for variously chosen values of Ai,ρ and pn, we heuristically

observe that in 99% of the trials, the value of the random variable lies within three

standard deviations of the mean.

In order for separation between estimated influence fields of object types to be

maintained whp, we require the following inequality to hold for each pair (i, i+1) of

objects.

E(di+1)−(3×√

V (di+1)) > E(di)+(3×√

(V (di)) (4.5)

Let ρfni(i+1)be the minimum density required to distinguish between objects i and

i + 1 whp. By solving Eqns. 4.3, 4.4, and 4.5, we obtain ρfni(i+1)as:

ρfni(i+1)= B

(pn+B×pn)

where

B = 9 × (√

A(i+1)+√

Ai)2

(A(i+1)−Ai)2

Theorem 4.2.1. The minimum network density, hatρfn, required to maintain sep-

aration between the estimated influence fields of all objects in the presence of false

negatives is the maximum of all pairwise densities ρfni(i+1).

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False positives

We now study the impact of false positives on estimation. In order for separation

to be maintained between the estimated influence fields of objects i and i + 1, we

require that the number of detections in area Ai, when combined with the false

positives in the region of area Ai+1−Ai, should be lower than the number of detections

in area Ai+1. The mean and variance of the number of false positives in the area

Ai+1 − Ai, respectively E(fpi) and V (fpi), are as follows.

E(fpi) = ρi(i+1) × (Ai+1 − Ai) × pfp (4.6)

V (fpi) = E(fpi) × (1−ρi(i+1) × pfp) (4.7)

We thus require the following inequality to hold in order to maintain separation

between the estimated influence fields of two objects Ai and Ai+1.

E(ni)+(3×√

(V (ni))+E(fpi)+(3×√

V (fpi)) <

E(ni+1)−(3×√

V (ni+1)) (4.8)

Note that the term on the right hand side does not contain any expression for

false positives. This is because false positives outside the area Ai+1 would affect the

estimation of both objects equally. Let ρfpi(i+1)be the minimum density necessary to

distinguish between objects i and i + 1, obtained by solving Eqns. 4.6, 4.7 and 4.8.

Theorem 4.2.2. The minimum network density, ρfp, required to maintain separation

between the estimated influence fields of all objects in the presence of false positives

is the maximum of all pairwise densities ρfpi(i+1).

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Density compensation techniques

The conditions derived above state the minimum density required for preserving

separation between the estimated influence fields of objects. However, in many cases,

network density is a function of other factors such as cost and communication range,

and thus may not be a parameter of choice. We therefore present techniques for

dealing with both inadequate and excess network density.

Temporal Aggregation: This technique is used to obtain the necessary separation

of estimated influence fields when the network density does not meet the minimum

requirements. Temporal Aggregation involves the following steps.

1. Using Theorems 1 and 2, compute minimum density ρ required to distinguish

all objects

2. Given network density ρ, choose the aggregation interval t such that ρ × t > ρ

3. Aggregate node detections over time t to estimate object influence fields.

The aggregated influence field, used in a spatio-temporal context, is the area

covered by an object in time t and it depends on the size, shape and motion model

of the object. The proof of why temporal aggregation helps achieve separation can

be deduced by rewriting Eq. 4.5 as follows.

E(di+1)−E(di) > (3×√

V (di+1))+(3×√

(V (di)) (4.9)

We see that when aggregated over time, expected values grow faster than standard

deviations, hence the desired inequality can be satisfied. Thus, temporal aggregation

can also be used to distinguish between objects that have the same influence field but

different speeds or motion models.

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Theorem 4.2.3. Separation between estimated influence fields of objects is preserved

in networks with insufficient density or for objects with different speeds, by aggregating

detections over interval t.

Probabilistic Reporting:

This technique is used to improve system efficiency and lifetime in cases where

network density exceeds the minimum specified by Theorems 1 and 2. Probabilistic

Reporting involves the following steps.

For each node:

1. Compute probability pr of reporting a detection.

2. For each detected object, send message to aggregator with probability pr

Computing pr: Consider the analysis presented earlier wherein each node detects an

object with probability pn. If each detecting node reports with probability pr, the

number of reporting nodes in an area Ai is a random variable with mean and variance

as given below.

E(ri) = ρ × Ai × pn × pr (4.10)

V (ri) = Ai × ρ × pn × pr × (1−ρ× pn × pr) (4.11)

In order for separation between estimated influence fields to be maintained whp,

the following inequality must hold for each pair (i, i+1) where (pr)i(i+1) is the proba-

bility of reporting.

E(r(i+1))−(3×√

(V (r(i+1))) > E(ri)+(3×√

V (ri)) (4.12)

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Let (pr)i(i+1) be the minimum probability required to distinguish between objects

i and i+1. By solving Eqns. 4.10, 4.11 and 4.12, we get the following.

(pr)i(i+1) = B((pn×ρ)+(B×pn×ρ))

where

B = 9 × (√

A(i+1)+√

Ai)2

(A(i+1)−Ai)2

Theorem 4.2.4. The minimum probability, pr, of reporting a detection required to

distinguish between all object types is the maximum of all pairwise reporting probabil-

ities (pr)i(i+1).

4.2.2 Network faults

In this subsection, we discuss the impact of fading and contention faults on reliable

estimation of influence fields.

Channel fading

Recall that pf is the probability of message reception over a single hop in the

presence of fading. Thus, over h hops, the probability of successful reception of

a message equals (pf)h. Since messages from nodes closer to the aggregator have

lower probability of failure, the estimated influence field of a small object close to

the aggregator can overlap with that of a large object far away. We therefore need

to compensate for the effect of distance of an object from the aggregator during its

estimation.

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Implementing distance insensitivity

We now present necessary conditions to maintain separation between the esti-

mated influence fields of object types in the presence of fading over multiple hops en

route to the aggregator. For simplicity, we assume that the object size is small as

compared to the distance from the aggregator, hence all detections corresponding to

the same object travel the same number of hops.

Probabilistic Reporting. To compensate for non-uniform reception probability, we

use the probabilistic reporting technique presented earlier. In this case however, the

reporting probability pr is not uniform for all nodes, rather it depends on the distance

to the aggregator.

Let D be the maximum number of hops to the aggregator in the network . The

probability of reporting for a sensor at distance of h hops from the aggregator is

chosen to be p(D−h)f . Thus, the probability of reporting for nodes D hops from the

aggregator is 1, while for h = 1, the probability of reporting is p(D−1)f .

We first show that distance-dependent probabilistic reporting compensates for the

effect of distance on estimation. Since fading errors are independent, the number of

successful transmissions at any distance has a binomial distribution. The number

of messages successfully received for an object i which is h hops away from the ag-

gregator, is thus a random variable f(h) whose mean and variance are obtained as

follows.

E(f(h)i) = Ai × ρ × pD−hf × ph

f = Ai × ρ × pDf (4.13)

V (f(h)i) = Ai × ρ × pDf × (1 − (ρ × pD

f )) (4.14)

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From Eq. 4.13 and Eq. 4.14, we observe that the distribution of the number of

successfully received messages is now independent of the number of hops.

We now derive a necessary condition for distinguishing objects whp. To achieve

this, it can be shown that for each pair of objects (i, i+1), the number of messages

successfully received whp for the smaller object i located at h = 1 should be less than

the number of messages successfully received whp for the larger object i+1 located

at h = D, as this represents the worst case. We thus have the following equation.

E(f(D)j)−(3√

V (f(D)j)) > E(f(1)i)+(3√

V (f(1)i)) (4.15)

Let ρi(i+1) be the minimum density required to maintain separation between esti-

mated influence fields of objects i and i+1. Solving Eqns. 4.13, 4.14 and 4.15, we

get the following.

ρi(i+1) = B(pD

f+(B×pD

f))

where

B = 9 × (√

A(i+1)+√

Ai)2

(A(i+1)−Ai)2

Theorem 4.2.5. Let each node h hops from the aggregator report its detections with

probability p(D−h)f where D is the maximum number of hops to the aggregator. The

minimum density ρ required to maintain separation between the estimated influence

fields for all object types in the presence of multi-hop fading faults, is the maximum

of all pairwise densities ρi(i+1).

Note that since our techniques are compositional, we can deal with conditions

where we have less than or more than this density ρ by using the techniques discussed

earlier.

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Spatial Reconstruction. As an alternative to probabilistic reporting, we present

the spatial reconstruction technique which involves the following steps:

1. Upon detecting an object, each node sends a message to the aggregator.

2. For each distance h, aggregator scales number of received messages from that

distance by 1ph .

Thus, if the aggregator receives k messages from distance h, it considers this as

having received kph messages. The scaling factor is a result of the uniform probability

of receiving a message from a node h hops away being phf . Spatial Reconstruction

is the dual of distance-dependent Probabilistic Reporting. In this case, all detecting

nodes transmit with the same probability, which may be lower than 1 for reasons

of efficiency. The minimum network density required to distinguish between object

types is the same as in Theorem 5 because the probability of fading loss is independent

of the number of messages being transmitted.

Channel contention

In this subsection, we analyze the effect of interference due to channel contention

on a single hop. In our event based traffic model, all nodes detect an object and

hence compete for the channel at nearly the same instant. Thus, as the event size

increases, the message losses increase too. We analyze the effect of channel contention

on aggregation, under the assumption of the following one hop model.

Suppose n nodes, all within one hop of each other and the aggregator, want to

send a detection message to the aggregator. Each node randomly chooses one of c

time slots for transmitting the message. Let c be greater than n. If multiple nodes

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choose the same slot, their messages collide and all of them are lost. This models a

random backoff MAC scheme, commonly used in wireless communications.

The expected number of messages successfully received by the aggregator is the

expected number of time slots that are chosen by exactly one node. This is an

instance of a classical occupancy problem in combinatorics. The probability that a

slot is chosen by exactly one node is equal to the probability that all other nodes

choose different slots. This is the probability that a message does not get lost due to

channel contention, which we denoted as pc in Sec. 3.3.1.

pc = (1 − 1/c)(n−1) (4.16)

The number of nodes with a time slot for themselves, i.e., the number of messages

that do not get lost due to channel contention is a random variable having a binomial

distribution with parameters (n, pc). The mean and variance of the distribution,

denoted as E(s) and V (s), are as follows:

E(s) = n × pc = n × (1 − 1

c)(n−1) (4.17)

V (s) = n × pc × (1 − pc) (4.18)

From Eq. 4.17, it is seen that for a given c, as n increases, the expected number

of successful messages reaches a maximum and then starts decreasing.

Definition 14 (Inversion point). The inversion point of a network with respect to a

given observer is the number of senders for which the expected number of messages

received is maximum.

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The inversion point, denoted as ninv, obtained by solving for the maxima of

Eq. 4.17 is as follows.

ninv =1

ln(1 + 1c)

(4.19)

Due to inversion, the aggregator may receive fewer detection messages for a larger

object than it receives for a smaller object, hence the separation between the estimated

influence fields of the object types may not be preserved.

Experimental results.

(a) (b)

Figure 4.2: Inversion in a one hop network.

Fig. 4.2 shows the experimentally measured impact of increasing the number of

transmitters on the network reliability of the single hop model. This experiment was

performed using Mica2 motes running TinyOS [32], using globally synchronized time

to generate concurrent messages. The nodes were placed within one hop of each

other and of the aggregator and their transmit power was set to be high enough to

negate fading losses. The experimental results in Fig. 4.2 have been averaged over 50

trials for each of the traffic loads of size 2,5,10,20,30 and 40 sources. For these traffic

loads, not only does the reliability of the network decrease significantly as the number

of nodes increases as seen in Fig. 4.2(a), but it leads to the inversion effect seen in

149

Fig. 4.2(b). This inversion may cause an overlap between the number of messages

received at the aggregator implying that previously separable influence fields will no

longer be separable.

Implementing contention insensitivity

In this subsection, we describe two techniques to compensate for inversion effects

produced by network contention to preserve separation between estimated influence

fields.

Probabilistic Reporting. We use the same technique described earlier to com-

pensate for contention effects, however the constraints for choosing the reporting

probability pr differ as follows.

If each detecting node reports with probability pr, the number of reporting nodes

is a random variable with expected value and variance as follows.

E(ri) = Ai × ρ × pr (4.20)

V (ri) = Ai × ρ × pr × (1 − ρ × pr) (4.21)

Recall from Eq. 4.16 that the probability of a message being successfully received

for an object i is dependent on the number of reporting nodes, which itself is a random

variable. We make a simplifying, yet conservative, assumption that while the number

of reporting nodes for an object is a random variable, the probability of successful

reception is uniform and depends on the expected number of reporting nodes. This

assumption results in a smaller traffic load than the expected value being subjected

to larger contention than it would really experience. Similarly, larger traffic loads

150

are subjected to lower contention than actual. Consequently, the interval over which

the number of received messages is distributed subsumes the interval that would

be obtained in practice. Hence, the necessary conditions for maintaining separation

between object types, resulting from our assumption are conservative. We now have

for object i, the following equation.

pci= (1 − 1/c)(E(ri)−1) (4.22)

Using Eqns. 4.20 and 4.22, the number of messages that are successfully received

for this object is now a random variable whose mean and variance are given by the

following equation.

E(si) = Ai × ρ × pr × pci(4.23)

V (si) = Ai × ρ × pr × pci× (1 − (ρ × pr × pci

)) (4.24)

For separation between estimated influence fields to be maintained whp, we require

the following inequality to hold for each pair (i, i+1).

E(s(i+1)) − 3 ×√

V (s(i+1)) > E(si) + 3 ×√

V (si) (4.25)

Selecting pr: Solving the above inequality yields a quadratic whose solutions de-

note the minimum and maximum probabilities of reporting for which the two object

types can be distinguished. The following procedure can be used to select the prob-

ability of reporting such that the estimated influence fields for all object types are

separable.

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1. For each pair (i, i+1), where 1 ≤ i and i < k, using Eq. 4.23, 4.24 and 4.25, ob-

tain a range of probabilities given by the closed interval (min((pr)ij), max((pr)ij)).

2. Let (prmin, prmax) denote the intersection of all such ranges.

3. If the intersection is not empty, choose pr = prmin.

Theorem 4.2.6. Assume pr is the probability determined by the selection procedure.

Separation between estimated influence fields of all objects is achieved when each node

reports its detections with probability pr.

Note that there may be cases where the selection procedure returns an empty

intersection in Step 2. We now describe an additional algorithmic technique to deal

with such cases.

Temporal Segregation. If the procedure described above returns an empty range

of feasible reporting probabilities, it means that there exist objects a, b, c in order of

increasing influence field sizes such that max((pr)ab) < min((pr)bc). Thus, there exist

pairs for which eliminating inversion requires such a small probability of reporting

that other objects are no longer distinguishable. To overcome this problem, the

inversion point ninv, has to be increased. According to Eq. 4.19, this can be achieved

by increasing the number of time slots c. In other words, we temporally segregate

the messages. Temporal segregation is also achieved by using additional application

level backoffs before reporting a detection. Note that one can also eliminate the

problem of channel contention by precisely scheduling the transmission of messages.

One example of such a scheme is TDMA. The drawback of all such schemes is that

they incur an additional delay overhead.

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Experimental results. Recall the inversion effect demonstrated experimentally in

Fig. 4.2. We now demonstrate how the techniques described above compensate for

the inversion effect, allowing us to distinguish between object types under consider-

ation. The experimental setup is the same as in the previous experiments with the

same internode distances, same transmit power and the same traffic source sizes.

(a) Probabilistic re-porting

(b) Temporal segre-gation

Figure 4.3: Dealing with inversion using Contention compensation techniques.

Fig. 4.3(a) demonstrates the results of using Probabilistic Reporting with proba-

bility 0.5. At each concurrent sending event, all the potential transmitters indepen-

dently decide whether or not to transmit their message. The graph shows that by

using probabilistic reporting, inversion is avoided for the traffic loads under consid-

eration. However, note that the intersection in this case is empty for sources of size

5, 10 and 20. Thus, even with probabilistic transmission, we do not achieve disjoint

ranges of message reception.

Fig. 4.3(b) demonstrates how Temporal Segregation helps achieve this separation.

In this experiment, the number of slots available for choosing when to transmit was

increased 4 times as compared to the previous case. As seen from the graph, by

increasing the number of slots, we are able to avoid inversion for the traffic loads

under consideration. The graphs also demonstrate that the overlap between messages

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received is eliminated for sources of size 5, 10 and 20. In fact, if the number of slots

were increased even further, there would be no overlap for any source size.

4.3 Putting it all together: Classification and Tracking

In this section, we show how the analysis and techniques for isolated fault classes

presented earlier are composed, and discuss the effect of end-to-end unreliability on

reliable estimation. We also present a procedure for distinguishing objects from false

positives and discuss how object locations can be tracked by shape estimation of

influence fields.

4.3.1 Composing fault classes

In this subsection, we describe how the necessary conditions for maintaining sep-

aration between estimated influence fields of objects can be derived when multiple

faults can occur simultaneously. We illustrate this compositional approach through

an example of node faults; other fault classes can be dealt with similarly.

Recall from Section 4.2, Eqns. 4.5 and 4.8, which specify conditions for main-

taining separation between the estimated influence fields of two objects i and i+1 in

the presence of false negatives and false positives respectively. However, if both these

faults can occur, neither of these conditions is sufficient. The necessary condition for

distinguishing between two objects i and i + 1 in the presence of both type of node

faults can thus be stated using the following equations.

E(di)+(3×√

(V (di))+E(fpi)+(3×√

V (fpi)) <

E(di+1)−(3×√

V (di+1)) (4.26)

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Let ρnodei(i+1)be the minimum density required to distinguish between i and i+1,

obtained using Eq. 4.26.

Theorem 4.3.1. The minimum density ρnode required to distinguish between all ob-

jects in the presence of both false negatives and false positives is the maximum of all

pairwise densities ρnodei(i+1).

4.3.2 End-to-end reliability

The method described above can be used to analyze the effects of multiple fault

classes affecting reliable estimation. We now present a unified analysis for studying

the impact of end-to-end reliability. This net reliability encapsulates all losses that

may be encountered including false negatives, fading and contention losses. We do

distinguish false positives in this analysis as they are additive faults. As discussed in

Sec. 3.3.1 and as we will demonstrate in the next section, we characterize end-to-end

reliability empirically. We denote the uniform probability of receiving a detection

from a node in the influence field as prcvih. Using distance-dependent Probabilistic

Reporting or Spatial Reconstruction, we can compensate for the effect of distance

on reliability. The mean and variance of the number of detections received by the

aggregator for object i, respectively E(rcvi) and V (rcvi), then are as follows:

E(rcvi) = ρ × Ai × prcvi(4.27)

V (rcvi) = Ai × ρ × prcvi× (1−ρ× prcvi

) (4.28)

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Similarly, the mean and variance of the number of false positives in the area

Ai+1 − Ai which are received by the aggregator, respectively E(fpi) and V (fpi), are

as follows.

E(fpi) = ρi(i+1) × (Ai+1 − Ai) × pfp × prcvi(4.29)

V (fpi) = E(fpi) × (1−ρi(i+1) × pfp × prcvi) (4.30)

From Eqns. 4.27, 4.28, 4.29 and 4.30, we obtain the following equations.

E(rcvi)+3×√

V (rcvi)+Efpi+3×√

V (fpi) <

E(rcvi+1)−3×√

V (rcvi+1) (4.31)

Let ρneti(i+1)be the minimum density required to maintain separation between the

estimated influence fields of objects i and i + 1, obtained using Eq. 4.31.

Theorem 4.3.2. The minimum network density ρnet required to maintain separa-

tion between all object types whp in the presence of false positives and end-to-end

unreliability in the network is the maximum of all pairwise densities ρneti(i+1).

4.3.3 Identifying and isolating objects

The analysis presented earlier describes how to choose network density and/or

fault compensation techniques so that the estimated influence fields of all objects

can be distinguished. Assuming these requirements are satisfied, we now describe a

procedure to filter false positives from a set of detections received for one or more

objects so that these objects can then be classified and tracked.

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Let ρnet be the network density which, as described in the previous subsection,

suffices to distinguish between all object types in the presence of faults. We now define

the active density adi for object i as the ratio of the minimum number of detection

messages that may be received for object i to the influence field area of object i.

adi =(E(rcvi) − 3 ×

√

V (rcvi))

Ai

(4.32)

Let admin be the minimum active density among all objects and let imin be the

object with this minimum active density. We then apply the following spatial filtering

algorithm to isolate object detections from false positives:

1. For each received detection m, apply filtering windows of size A1 around it.

2. If a window contains more than A1 x admin detections, mark all these as object

detections, else mark m as a false positive.

3. Repeat steps 1 and 2 till no more detections are unmarked.

It can be shown that the spatial filtering window size in Step 1 should be the

influence field area of the smallest object, A1. The uniform reliability property of the

routing protocol in our model guarantees that if the minimum active density threshold

is exceeded in a window, then all detections must belong to some object type. The

above procedure thus identifies and isolates object boundaries.

4.3.4 Tracking using shape estimation

Having derived the minimum network density and/or compensation technique to

achieve separation between estimated influence fields of detected objects, we can use

the estimated shape of the influence field to track the object location. For instance,

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a metallic object, which generates a uniform, circular influence field around it can be

tracked at the centroid of the locations of magnetometers that detect it, while a light

source producing a conical beam can be tracked at the vertex of the photosensors

detecting it.

To achieve tracking, we require that nodes from which detections used in shape

estimation are received, should be distributed uniformly across the influence field,

otherwise the estimated shape of the influence field may be distorted. False negatives

and false positives occur independently at nodes hence their distribution is uniform

across the influence field. The one-hop contention model for faults is based on nodes

randomly selecting the same slot for transmission, hence the distribution of these

faults is uniform. Also, recall from Sec. 3.3.1 that the GridRouting protocol has the

property of uniform reception probability for nodes equidistant from the aggregator.

In the multi-hop case though, the probability of failure is non-uniform because far-

ther nodes are subject to a higher loss rate. However, as shown in Section 4.2, the

techniques of distance dependent probabilistic reporting and spatial reconstruction

compensate for this non-uniformity of network failures.

4.4 Case study: A Line In The Sand

In this section, we describe the design and implementation of a distributed classifi-

cation and tracking system which we called A Line In The Sand. This system consists

of 90 Mica2 motes deployed in a 1.5m spaced grid to cover a 18m x 7m area. A Line

In The Sand has been deployed in several outdoor settings to accurately distinguish

between civilians, soldiers and vehicles by estimating their influence fields based on

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magnetometer and micro-power impulse radar sensors. For simplicity of presenta-

tion, we only describe classification between a soldier and a car using magnetometer

based influence fields. We first describe how we derived system parameters like den-

sity using the theorems presented earlier. We then validate, both theoretically and

experimentally, that accurate classification and tracking can be achieved whp (99%)

in this network.

4.4.1 Experimental measurements

We first describe the experimental setup to measure key system parameters. To

measure the magnetic influence fields of a soldiers and a car, a dense, regular grid

of Mica2 motes with magnetometers was deployed. These objects were then made

to traverse this network at different speeds and orientations. We then averaged the

observations from over 100 such trials. The influence field areas A1 and A2, for

a soldier and a car, were thus measured to be 12m2 and 63m2 respectively. The

probability of node faults, 1 − pn, which included node failures and false negatives

was measured to be 10%. By observing the number and distribution of false positives

in the network over time and location, we determined the probability of false positives,

pfp to be 2%.

4.4.2 Determining network density

Substituting for the experimentally measured values of A1, A2, pn and pfp in

Theorem 7, we obtained the following conditions for minimum network density:

ρ01 > 0.6 , ρ12 > 0.65

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where ρ01 is the minimum density needed to distinguish false positives from a

soldier and ρ12 is the minimum density needed to distinguish a soldier from a car.

The minimum density needed to distinguish between all three, ρ was thus 0.65. Based

on the communication radius of these nodes, we deployed 90 nodes in a 1.5m spaced

grid to cover the 18m x 7m area with a network density of 0.7.

4.4.3 Effect of network unreliability

As mentioned in Section 4.3, we characterized the end-to-end network reliability

function empirically. To do so, we deployed the network with the desired density of

0.7 and repeated the earlier trials with a soldier and car traversing the network and

measured the network reliability at the aggregator. By averaging over more than 100

such trials, we obtained the values of prcv1 and prcv2 as 0.85 and 0.55 respectively.

Substituting for prcv1,prcv2 and pfp in Eqns. 4.27, 4.28, 4.29 and 4.30, we obtain the

following equations.

E(rcv1) = 7.1, V (rcv1) = 2.9

E(rcv2) = 24.25, V (rcv2) = 14.9

E(fp1) = 0.6, V (fp1) = 0.6

It can be seen that these values satisfy the inequality in Eq. 4.31, thereby validating

that the estimated influence fields of a soldier and a car can be separated in the

presence of end-to-end unreliability and false positives.

4.4.4 Experimental validation

We now present experimental data to demonstrate that we were able to distinguish

between the given object types in the presence of node faults and network unreliability.

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(a) Measuredat motes

(b) Measuredat aggregator

(c) Temporalevolution

Figure 4.4: Impact of network reliability on influence fields in A Line In The Sand

Fig. 4.4(a) shows the probability distribution function for the influence fields of a

soldier and a car as measured at the aggregator. In this outdoor experiment, times-

tamped detections were recorded in the non-volatile memory at each mote during 100

runs of a soldier and vehicle each, moving through the network. These detections

were then downloaded and time-correlated to recreate the measured influence fields

and their probability distribution. It can be seen from Fig. 4.4(a) that these mea-

sured influence fields are indeed clearly separable. Fig. 4.4(b) shows the probability

distribution of the influence fields estimated by the aggregator based on detections

received using the GridRouting protocol in 100 runs of a soldier and a car each. It

can be seen that in the presence of false positives and network unreliability, sepa-

ration between the estimated influence fields of a soldier and a car is lower than in

Fig. 4.4(a). However, as calculated earlier, we see that these distributions are non-

overlapping meaning that the number of detections received for a vehicle is always

greater than that for a soldier. From this data, we also calculate that there exists

little variability (7.2%) in network reliability across these runs. Fig. 4.4(c) shows the

temporal evolution of influence field estimation at the aggregator. The proximity of

traces for individual runs indicates that in addition to reliability, network delays are

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also quite predictable, allowing us to choose tight latency bounds at the aggregator.

This serves to validate our model that uniform, predictable end-to-end reliability can

be obtained by careful design of the routing protocol.

Figure 4.5: Classification and tracking of a car in A Line In The Sand

4.4.5 System performance

Finally, we give some performance data for A Line In The Sand. By considering

the influence field analysis and appropriately tuning the desired network parameters,

we were able to achieve the desired classification accuracy of 99%. The accuracy of

tracking was higher for a soldier (1-2m) as compared to a vehicle (3-5m) providing

further evidence of the claim that reliability and uniformity are dependent on the

object type. The system was able to classify and track multiple objects moving

concurrently through the network as long as they were separated by a minimum

distance threshold. Fig. 4.5 shows a snapshot of the classification and tracking output

produced by the system for a car moving through the network.

4.5 Extensions to the Influence Field Approach

In this section, we describe extensions to the influence field to increase the con-

fidence in classification and also decrease the required density of deployment. One

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such extension is the use of multiple sensing modalities at each node to estimate mul-

tiple influence fields. As an example, consider the case where we wish to distinguish

between a motorcycle, a SUV and a truck. The influence field areas for these objects

using a magnetometer are 28m2, 78m2 and 150m2 respectively. The minimum den-

sity required to distinguish all three objects in the presence of faults, as calculated

using the theorems presented earlier, is quite high (2-3m grid spacing). However,

the influence field areas of the same objects with respect to an acoustic sensor are

1250m2, 700m2 and 1250m2 respectively. By estimating the influence fields for both

these modalities, it is possible to distinguish between these objects with a much lower

network density (8-10m grid spacing). The basic idea is to use the estimated acoustic

influence field to distinguish a SUV from the other two object types and then use

the magnetic influence field to distinguish between a motorcycle and a truck. Thus,

using multiple sensing modalities reduces the network density to 6-10% of what was

originally required. This simple example serves to demonstrate that estimating mul-

tiple influence fields can be used to classify multiple types of objects with higher

confidence and lower network density. Multimodal influence field estimation was suc-

cessfully demonstrated in the ExScal[4] network, which is one of the largest sensor

deployments to date. ExScal used three types of sensors – magnetometers, acoustic

and motion sensors – to estimate the influence fields of a person, a car and an ATV

and classify and track them accurately.

Another extension to the influence field concept involves communicating not only

presence outputs from each node, but some vector of features about the object like

peak amplitude, total energy or distance from the object. Such an enhanced influence

field can be used to improve confidence in the estimation output.

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Regardless of whether the basis chosen for determining the influence field is merely

presence as in the case of A Line In The Sand, a combination of multiple modalities,

or a vector of features, reliable estimation is still an important problem and the same

techniques described in this paper can be applied to each of these extensions.

4.6 Related Work

The instrumentation of a militarized zone with distributed sensors is a decades-

old idea. Unattended ground sensors (UGS) exist today that can detect, classify,

and determine the direction of movement of intruding personnel and vehicles. The

Remotely Monitored Battlefield Sensor System (REMBASS) exemplifies UGS systems

in use today. REMBASS exploits remotely monitored sensors, hand-emplaced along

likely enemy avenues of approach. These sensors respond to seismic-acoustic energy,

infrared energy, and magnetic field changes to detect enemy activities. REMBASS

processes the sensor data locally and outputs detection and classification information

wirelessly, either directly or through radio repeaters, to the sensor monitoring set

(SMS). Messages are demodulated, decoded, displayed, and recorded to provide a

time-phased record of intruder activity at the SMS.

Like REMBASS, most of the existing radio-based unattended ground sensor sys-

tems have limited networking ability and communicate their sensor readings or in-

trusion detections over relatively long and frequently uni-directional radio links to a

central monitoring station, perhaps via one or more simple repeater stations. Since

these systems employ long communication links, they expend precious energy during

transmission, which in turn reduces their lifetime. For example, a REMBASS sensor

node, once emplaced, can be unattended for only 30 days.

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In contrast, we design a dense, distributed, and 2-dimensional sensor network-

based classification and tracking system using inexpensive sensor nodes. In this

model, intrusion data are processed locally at each node, shared with neighboring

nodes if an anomaly is detected, and communicated to an exfiltration gateway with

wide area networking capability.

The notion of influence of an energy source is used in other science and engineering

applications. In some formulations, the distribution of the intensity of the source at

various points is considered while modelling its influence. For example, Kellogg et

al [6] model the temperature distribution of a heat source across a region as an

influence graph and use the graph to design algorithms for distributed control. In

other formulations, including Zhao et al [74] and ours, the distribution of the intensity

is not modelled. Zhao et al [74] define an influence area as the number of sensors

that detect an object. Our definition of the influence field also captures the shape

of the influence field. We are unaware of previous work that has used influence field

estimation as a basis for classification and tracking of objects.

The influence field approach should be contrasted to traditional approaches for

classification and tracking using Unattended Ground Sensors [31]. The Remotely

Monitored Battlefield Sensor System (REMBASS) is a representative example. The

approach is centralized, requires complex pattern matching[16] and the sensing and

processing devices are expensive, require careful and precise deployment as well as fre-

quent remote monitoring. Meesookho, et al [55] describe a collaborative classification

scheme based on exchanging local feature vectors, which imposes a high load on the

network. By way of contrast, most of the work on distributed tracking that decreases

the load on the network is based on collaborative signal and information processing,

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sequential Bayesian filtering, and extended Kalman filtering [74, 21, 50, 51, 54, 75, 26],

that require significant node computation.

For the case of node faults, Krishnamachari et al [45] have presented probabilistic

decoding mechanisms to detect regions of events in the presence of uncorrelated sensor

faults with relatively low probability (around 10%). Our work accommodates the case

of uniform nodal failures and we have also presented techniques to handle network

faults whose impact is non-uniform across the network such as fading, and network

faults whose failure probability grows with the event size such as contention. To the

best of our knowledge, the impact of network unreliability in estimating the influence

field has not been addressed before, nor has it been addressed in the context of

distributed classification and tracking, which has led us to the present work.

Our work also relates influence field to sensor coverage [29, 73] and highlights the

important similarities and distinctions between the two concepts. Existing work on

sensor coverage gives necessary conditions for detecting an object moving through a

network. We are not aware of extensions that deal with the problem of classification.

4.7 Summary

In this chapter, we considered the problem of reliably estimating the influence

fields of different target types in a wireless sensor network subject to a variety of

faults. We provided mechanical procedures for sensor node density selection as well

as algorithmic techniques appropriate for dealing with each fault class. Corroboration

of our results and techniques was provided through at-scale experiments.

We showed how reliable estimation was achieved to enable accurate classification

and tracking in A Line In The Sand. The case study also provided a data point for

166

the significant impact of network unreliability on network and application design, as

well as one for a need for routing protocols in sensor networks to provide uniform

reliability.

Our work reveals a notable co-dependence between application design and network

design. To achieve the desired estimation reliability, we needed in some cases to

use both techniques that affected the network (such as tuning of MAC or routing

protocol parameters) and that affected the application (such as tuning the probability

of reporting and the rate of temporal aggregation). How to design stable and scalable

systems when there are such cyclic dependencies involved is an issue of interest to us.

Although our compositional models allow us to reason about the effects of different

types of node and network faults, there are some relevant and more complex fault

models that we have not dealt with analytically. One such model, which we dealt

with only experimentally in A Line In The Sand concerns multi-hop contention and

fading errors. In future work, we seek to address this model analytically. We will

also incorporate in our analysis consideration of multiple concurrent targets that we

dealt with experimentally, towards addressing the gap between existing theory and

practice.

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CHAPTER 5

DISTRIBUTED VIBRATION CONTROL

Sensor-actuator networks are being prototyped in the control of distributed pa-

rameter systems such as flexible structures. A specific example is the vibration control

of a fairing during payload launch using embedded MEMS components based sensor-

actuator networks [12, 72]. Since MEMS based sensor-actuator devices are potentially

cheap, a large number of these devices can be embedded on flexible structures and

combinations of these sensors can be used to obtain the required mode vibration

information and then the output from these combinations can be used to provide

adequate distributed control. Similar applications arise in the control of chemical

plants and nuclear reactors.

Distributed control systems have applications in space missions and nuclear plants

where degradation of system performance may even compromise human safety. Hence

satisfactory performance in the presence of faults is a requirement for these systems.

The constraint in most distributed control applications is that of mission critical

stability, but achieving this using wireless sensor/actuator networks is a challenge.

Sensor-actuator network based control systems typically comprise of embedded sen-

sors and actuators, microprocessor-based controllers (central or distributed) and an

underlying network that provides information processing services to the controllers

168

such as controller group synchronization, communication, (re)parameterization, re-

configuration, etc. Each of the above subsystems are subject to faults: there are

hardware faults and these will increase when subject to harsh and unpredictable en-

vironments, there are faults in the underlying software and middleware services such

as information loss, delay and corruption, and there are configuration faults which

given the scale of these networks this will increase even more.

We performed a series of experiments to analyze the effect of potential faults on

a vibration control system. We carried out our experiments on the Boeing Open

Experimental Platform, which is a simulation framework intended to capture the

vibroacoustic damping problem on a satellite launch vehicle.

Figure 5.1: Fairing Shaped Payload Installed with Sensors and Actuators

In Fig. 5.1 we have shown a fairing shaped payload that is installed with sensors

and actuators on its surface that is used to control the vibrations during payload

launch. The simulated environment of the Boeing OEP application includes a fairing

169

plant model which is a simulation of the fairing structure, a hierarchical control ap-

plication [33, 34] and a fault injection framework. The 100 node system is partitioned

into several groups. Each group acts to damp a particular mode of vibration in the

fairing. The number of nodes assigned to each mode depends on the frequency of

that mode and the energy required to achieve the dampening. There can be as many

as 20 modes to address.

We identified potential component and network level faults for the fairing control

application [43]. Using a fault injection framework we evaluated the effect of these

faults on the control performance of the Boeing OEP. Specifically, the fault types that

we considered are: node fail-stop, node crash, nodes behaving randomly (in terms of

sensing and actuation), nodes debonding from their surface and network faults.

From our experiments [43], we note that the underlying hardware faults can result

in arbitrary behavior of sensor and actuators that can cause substantial degradation of

performance. Our observations motivate the need for designing reliable control scheme

that maintain stability and performance in the presence of arbitrary component faults.

One of the methodologies for the design of fault-tolerant control systems involves real-

time fault detection, isolation and control system reconfiguration [11, 39, 61, 44, 13].

An appropriate action is taken after the diagnosis of the faults. This method still

leaves the following challenges. The hardware itself can be faulty causing the actuators

to fail-stop and offer no control or debond from their surface causing them to offer

incorrect control. It is sometimes also not feasible to integrate the fault detection,

diagnosis and reconfiguration in dynamical systems particularly when the available

reaction time is limited. The underlying fault detection service is itself vulnerable to

faults in the middleware and software services. This leads us to consider a Byzantine

170

model for the actuator faults. A Byzantine actuator can produce an arbitrary control

input to the plant at all times. The behavior is non-deterministic and it can even

be the worst possible value at all times. Assuming this worst case scenario, we focus

on designing systems that maintain asymptotic stability in the presence of Byzantine

actuators that apply arbitrary control input to the plant.

Problem statement

Assuming that a bounded number of network actuators can exhibit incor-

rect (and potentially arbitrary) behavior, how can distributed control be

designed to be provably stable?

In this chapter, we model the system S to be marginally stable, linear time-

invariant and multi-variable with m sensor-actuator pairs. We apply distributed local

output feedback control to stabilize the system. The question then arises as to how

can the control system be guaranteed to be stable when a fraction of the actuators

in the network are Byzantine. Byzantine insensitivity implies that stability of the

system is guaranteed despite a fraction of the actuators being Byzantine.

Given a maximum of k Byzantine actuators, we first determine a necessary number

of actuators to guarantee asymptotic stability. Then for a 2 dimensional system, we

determine a sufficient number of redundant actuators and determine conditions on

placement of the actuators that will guarantee asymptotic stability of the control

system. We demonstrate our approach using a beam vibration control application as

a case study.

In Section 5.1 we describe the system and fault model and provide a sufficient

condition for the stability of the system in the absence of faults. In Section 5.2, we

first design a reliable control scheme using redundant colocated actuators and then

design a reliable control scheme where the redundant actuators are not colocated and

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the redundancy is further decreased. In Section 5.3, we demonstrate our methodology

using a beam vibration control application [7, 64] as a case study. In Section 5.4, we

discuss related work. We present a summry in Section 5.5.

5.1 System and Fault Model

In this section we describe the system and fault model and derive sufficient con-

ditions for the asymptotic stability of the system without faults.

5.1.1 System Model

Consider a marginally stable linear time-invariant multivariable system S with m

sensor-actuator pairs, described by the following equations and control law.

x = Ax + Bu (5.1)

y = Cx (5.2)

where x is an n-dimensional state vector [x1, x2, · · · , xn]T , u is an m-dimensional

actuator vector, B is an n×m dimensional matrix and the individual sensor-actuator

pairs are colocated. We assume that the system is controllable and observable from

individual locations. Since S is marginally stable, A has eigenvalues on the imaginary

axis. Since the individual pairs of sensors and actuators are colocated, we have the

following condition.

B = CT (5.3)

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Starting at any state, without any control being applied the system maintains its

energy as it is marginally stable. We apply the following local on-off output feedback

control law to stabilize the system.

ui = α × sign(yi), i = 1....m (5.4)

where α is less than zero.

Further ui equals zero when yi is zero. Thus a correct actuator can have 3 possible

control values 0, −α and α. We choose |α|, the magnitude of the actuator force, to

be the maximum force that an actuator can apply and assume that this is the same

across all actuators.

5.1.2 Asymptotic Stability Without Faults

We now analyze and prove the stability properties of S in the absence of faults.

Theorem 5.1.1. If m ≥ n and the matrix B is of rank n, the system S is asymptot-

ically stable.

Proof. We use the Lyapunov approach to prove stability. Now, let us define function

V as

V = xT Mx (5.5)

where M is a symmetric, positive definite n × n matrix. The Lyapunov derivative

can then be written as

V = xT (AT M + MA)x + 2xT MBu (5.6)

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Since A is marginally stable, we can transform A to be skew symmetric and AT +A

equals zero. Thus M can be the identity matrix.

V = 2xT Bu (5.7)

Let Bi denote the ith column of matrix B. For the system described in Eq. 5.1,

we have

V = 2 × α(m

∑

i=1

(xT ).(Bi) × sign(yi)) (5.8)

= 2 × α(

m∑

i=1

(xT ).(CTi ) × sign(yi)) (5.9)

= 2 × α(m

∑

i=1

(yi) × sign(yi)) (5.10)

= 2 × α(

m∑

i=1

|(xT ).(Bi)|) (5.11)

Note that we can use the magnitude of the dot product (xT ).(Bi) because we see

from Eq. 5.10 that (yi)×sign(yi) is always positive. Since m is at least equal to n and

B is of rank n, the state x can be orthogonal to at most n − 1 actuators. Hence the

Lyapunov derivative is strictly negative. Thus the system is asymptotically stable.

5.1.3 Fault Model

We now describe the fault model acting on system S. We start with the definition of

a Byzantine actuator.

Definition 15 (Byzantine actuator). A Byzantine actuator q is one that can generate

arbitrary value of uq in the range −α to α at all times.

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We note that a Byzantine actuator behavior also captures the case of an actuator

fail-stopping (uq = 0, and an actuator debonding form its surface thereby applying

a fraction of the control force (0 ≤ uq ≤ α). In our fault model, k out of the m

actuators are Byzantine in system S.

We will prove that the system remains asymptotically stable even when the Byzan-

tine actuators behave in the worst possible way at all times. This is described below.

Let uci(t) be the correct actuator value at any time t for actuator i. Let ufi(t) be

the corresponding value generated if the actuator is Byzantine. We then have the

following conditions.

W1 : uci(t) 6= 0 ⇒ ufi(t) = −uci(t) (5.12)

W2 : uci(t) = 0 ⇒ ufi(t) = ±α (5.13)

Note: If the system S is in equilibrium and is acted upon by a Byzantine actuator,

then the system is subject to perturbation and the energy of the system increases.

We do not consider this case in our fault model. We are interested in maintaining

the asymptotic stability of S in the presence of Byzantine actuators.

5.2 Reliable Control System Design

In this section, we design two reliable control schemes that maintain asymptotic

stability of the System S in the presence of Byzantine actuators.

175

5.2.1 Reliable Control System Using Redundant ColocatedActuators

In this scheme we place multiple actuators at each location. Thus the effect of

each redundant actuator on the control stays the same.

Theorem 5.2.1. A sufficient condition to tolerate k Byzantine actuators at each

location and guarantee asymptotic stability in the system S is to have 2k+1 actuators

at each of the m locations, where m ≥ n and the B matrix formed by the m distinct

locations is of rank n.

Proof. Since there are 2k + 1 actuators at each location, the Lyapunov derivative in

Eq. 5.11 can be written as follows

V = 2 × α(

m∑

i=1

((2k + 1) × |(xT ).(Bi)|)) (5.14)

First of all, we see from Eq. 5.14 that if the actuators are not Byzantine, the

redundant actuators still keep the energy derivative negative. We now analyze the

effect of Byzantine actuators at each location. Without loss of generality let us

consider the qth location and assume that k actuators at this location are Byzantine.

We consider the 2 conditions W1 and W2, described in the fault model.

When condition W1 of the fault model applies, the energy derivative term corre-

sponding to the qth actuator location can be written as follows.

Vq = 2 × α((k + 1) × |(xT ).(Bq)| − (k) × |(xT ).(Bq)|) (5.15)

= 2 × α(|(xT ).(Bq)|) (5.16)

176

Thus we see that the energy derivative corresponding to the qth location still stays

negative. This can similarly proved for all locations.

Now consider condition W2. If ucq(t) = 0, it implies that yq(t) = 0, i.e the local

output is zero. Thus the current state x(t) is orthogonal to the vector Cq. Since the

actuators are colocated, the current state x(t) is also orthogonal to the vector Bq.

Thus, the term xT .(Bq) is equal to zero no matter what force the Byzantine actuator

applies.

Hence the system S with 2k + 1 actuators at each of the m locations, is asymp-

totically stable in the presence of k Byzantine actuators at each location.

Note that this scheme tolerates k Byzantine actuators per location. If the expected

reliability ratio of the actuators are known, then we can design for the number of

actuators required at each location.

Given a reliability ratio for the actuators (greater than 0.5), denoted as ρ, we can

choose a k such that the system is reliable against Byzantine faults.

k

(2k + 1)> (1 − ρ) (5.17)

Note However it should be pointed out that placing the redundant actuators at the

same location may not be feasible in all control systems. Moreover the redundancy

rapidly increases as n increases because the actuators are replicated at each location.

We now describe a reliable control scheme where the colocation of redundant

actuators is not required and given that k actuators are Byzantine we add redundant

controllers to the system as a whole thus decreasing the redundancy required.

177

5.2.2 Reliable Control System Without Using Colocated Ac-tuators

We first state the minimum number of actuators to be added to the system S

which ensures that the energy derivative of Eq. 5.11 is less than zero at all times.

Lemma 5.2.2. For the energy derivative of Eq. 5.11 to be less than zero at all times

in the presence of k Byzantine actuators, we require m >= 2k + n

Proof. Let the number of actuators in the system be 2k + n − 1. The state x can be

orthogonal to at most n − 1 actuators. Let all of these be non-Byzantine actuators.

Thus the energy derivative terms corresponding to these actuators is zero. There are

2k actuators left. Without loss of generality assume that in the presence of any k

Byzantine actuators belonging to set of 2k actuators, the energy derivative is less

than zero. Then for the same state x, if the remaining set of actuators had been

Byzantine the energy derivative would be greater than zero. Thus we need at least

2k +n actuators for the energy derivative of Eq. 5.11 to be less than zero at all times

in the presence of k Byzantine actuators.

However, finding an actuator configuration that satisfies such a lower bound for

any k and n is a complex problem. We now focus our attention on second order

systems, i.e n = 2 and show that 3k+1 is an upper bound on the number of actuators

needed.

Definition 16 (m-uniform configuration system). An m-uniform configuration of the

system is the actuator configuration of the system in which each of the m columns

of the B matrix has the same amplitude and are uniformly distributed in the state

178

space of n dimensions such that the column vectors of B are pairwise equi-angular

and the angle between consecutive pairs of vectors is equal to πm

.

A second order system with 4-uniform and 7-uniform configuration is depicted in

Fig. 5.2. For simplicity, Let α be equal to 1. Thus given the actuator locations, each

actuator vector can either be equal or opposite to the direction shown depending on

the current state of the system.

(b) 7−uniform configuration

1

U2

U3

U4

45o

45o

45o

xθx 0o

x 45o

U1

U2

U3U4U5

U6

U7

x 0o57

25ox

57

25o

C(U − U )1 4 C(U − U )

1 7

x’1

x’2

x’1

x’2

θ

the angles between two vectors are

(a) 4−uniform configuration

U

Figure 5.2: 4-uniform and 7-uniform configurations for the second-degree system

Let a unit state vector xθ form an angle θ with the vertical axis as shown in the

figure. When an actuator is behaving correctly, the actuator vector would be such

that its dot product with the state vector is less than or equal to zero. This is because

the system is observable from each location and each actuator applies control in a

direction opposite to that of the local output. The inner product would be equal to

zero when the actuator vector is orthogonal to the current state.

Thus, the 4-uniform configuration shown in Fig. 1(a) is the proper actuator config-

uration when θ is between 0 and 45. In this configuration, the four normal actuators

179

U1, · · · , U4 keep the energy derivative negative when θ lies anywhere between 0 and

45. Note that the dot product of the state vector with each actuator vector is less

than or equal to zero. If θ is between 45 and 90, the first actuator changes its direc-

tion so that U1 is the new actuator vector. Thus the whole configuration is rotated

by 45 in the clockwise direction. Thus in an m − uniform configuration, if all the

actuators are correct, then the actuator vectors remain pairwise equi-angular at all

times.

Therefore while showing that a particular m−uniform configuration is sufficient

to guarantee asymptotic stability in the presence of Byzantine faults, it is enough to

consider the case that the unit state vector xθ is located in the basic range [0.0, 45.0].

In general, the basic range of m-uniform configuration system is [0.0, πm

]. Also note

that it is enough to consider unit state vectors because all the actuator vectors are of

same magnitude and the total dot product depends only on the angle.

In an m-uniform system of second-degree(m = 3k+1), let S(k, θ) denote the set of

k Byzantine faulty actuators such that, for a unit state vector xθ, the corresponding

energy derivative becomes maximized among all possible k subsets of actuators. Let

ED(k, θ) be the corresponding energy derivative.

For example, in the 4-uniform configuration of the system, S(1, 0) and S(1, 45)

are U2 and U3, respectively.

ED(1, 0) = x0θ · (U1 − U2 + U3 + U4)

= (cos 135 − cos 180 + cos 235)

= −0.4142

180

Likewise, ED(1, 45) turns out to be equal to −0.4142. Thus, in the boundary

angles of the basic range [0.0, 45.0], the system is asymptotically stable due to the

negative values of ED(1, 0) and ED(1, 45).

It is seen that for any m-uniform configuration, when the state vector is at the

boundary of the basic range, one of the actuator vectors is orthogonal to the state

vector and offers no control. Thus if ED(k, 0) and ED(k, πm

) are both negative, the

system is asymptotically stable in the presence of k Byzantine faults. We now write

down the expressions for ED(k, 0) and ED(k, πm

) in any m-uniform configuration.

φ = π/m = π/(3k + 1) (5.18)

ED(k, φ) =k

∑

i=1

cos(π

2+ i · φ) −

2k∑

i=k+1

cos(π

2+ i · φ) +

3k+1∑

i=2k+1

cos(π

2+ i · φ)

ED(k, 0) =

k−1∑

i=0

cos(π

2+ i · φ) −

2k−1∑

i=k

cos(π

2+ i · φ) +

3k∑

i=2k

cos(π

2+ i · φ)

ED(k) = min(ED(k, 0), ED(k.φ)) (5.19)

Upon numerical analysis of ED(k) for a large spectrum of values for k from 1

to 1000, it turns out to be that all values of ED(k) are negative as shown in the

figure below. Thus an m-uniform configuration of actuators is sufficient to guarantee

asymptotic stability of a second order system in the presence of k Byzantine actuators

when m = 3k + 1.

181

Figure 5.3: The maximum energy derivative ED(k) in m-uniform configuration system

Remark Note that the case of n = 2 and k = 1, where we need 4 actuators to

guarantee asymptotic stability satisfies the lower bound 2k + n. Further, an upper

bound on the redundancy required to tolerate k faults for higher dimension systems

can be found in a related technical report [47].

5.3 Application to Beam Vibration Control System

We now apply our reliable control system designs on a local output feedback

control scheme to a beam vibration control system. Given is a uniform beam of unit

length, unit mass, and unit stiffness factor, that is restricted by pins at both ends

and subjected to an initial disturbance. The beam has no dampening factor so that

it may vibrate endlessly. The beam has colocated velocity sensors and actuators to

reduce the vibration. For simplicity, we consider two fundamental modes of vibration.

The two fundamental vibration modes, denoted as M1 and M2, are derived [56]

as follows:

182

M1 : 1.4142 sinπz, λ1 = ω21 = 97.41 (5.20)

M2 : 1.4142 sin 2πz, λ2 = ω22 = 1558.55 (5.21)

where z ∈ [0.0, 1.0] denotes the position in the beam spatial axis and λi and ωi,

i = 1, 2, represent the eigenvalues and the frequencies of i-th modes, respectively.

Since each mode is governed by a second-degree differential equation, the state

vector for the system contains four variables x = [x1, x2, x3, x4]T . x1(x2) and x3(x4)

denote the vertical displacement and velocity of first (second) vibration mode, respec-

tively. Then, the system matrix A in Eq. 5.1 is denoted as

A =

0 0 1 00 0 0 1

−97.41 0 0 00 −1558.55 0 0

Note that we use a velocity feedback control. So the control input does not have

any effect on the states x1 and x2. The actuation is used to control the velocity states

x3 and x4. We will assume that the beam cannot be deformed permanently. Thus

when the velocity of the beam comes to zero, the displacement is also zero. Thus in

this specific example although the number of states is 4, the control affects only the

2 velocity states.

We first show that using 2 sensor-actuator pairs that form a B matrix of rank 2,

we can asymptotically stabilize the system. We choose the following B matrix.

B =

0 00 01 1.4142

1.3066 1

183

The Fig. 3(a) shows the energy of the system staring from an arbitrray initial

state going down to zero in the absence of faults. The energy of the system at time

t is calculated as xT (t) × X(t), where x(t) is the state of the system.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5800

850

900

950

1000

1050

1100

1150

1200

1250

1300

Time

Ene

rgy

(a) 1 actuator at each location, no faults

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5800

850

900

950

1000

1050

1100

1150

1200

1250

1300

Time

Ene

rgy

(b) 3 actuators at each location of which 1is Byzantine

Figure 5.4: Energy of the Beam Vibration System

We now show that 1 Byzantine actuator at each location can be tolerated and

asymptotic stability can be maintained by having 3 actuators at each location. The

Fig. 3(b)shows the energy of the system staring from an arbitrray initial state when

one actuator at each location is Byzantine.

We now show that when k = 1, we can asymptotically stabilize the system using

4 actuators that are distributed according to the 4 − uniform configuration. We

choose 4 pairs of colocated sensors and actuators such that the columns of B matrix

have equal magnitude and successive column vectors are seperated by an angle π4.

184

B =

0 0 0 00 0 0 0

−0.5754 0.1715 0.8179 0.9852−0.8179 −0.9852 0.5754 0.1714

The following graphs show the states of the 4 − uniform configuration system

staying asymptotically stable in the presence of no actuator faults and one actuator

failing.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5800

850

900

950

1000

1050

1100

1150

1200

1250

1300

Time

Ene

rgy

(a) Energy (4-uniform configuration) withNo Faults

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5800

850

900

950

1000

1050

1100

1150

1200

1250

1300

Time

Ene

rgy

(b) Energy (4-uniform configuration) with 1Byzantine Actuator

Figure 5.5: Energy of the 4-Uniform Configuration Beam Vibration System

5.4 Related Work

The application of active control to attenuate lower frequencies of vibration in

structures has been studied from many years [56, 8]. The earlier systems were mostly

centralized in control. However, the extension of this technology to large scale systems

motivates the design of control that is distributed [19].

185

A control system designed to tolerate failures in system components while main-

taining closed loop system stability and performance has been defined as a reliable

control system [71]. Such systems are also called systems possessing integrity against

component failures. Redundancy is a key ingredient in all such reliable control sys-

tems. A basic difference between robust control techniques and reliable control is

that the former deals with small parameter variations and system model uncertain-

ties while the latter handles more drastic changes in the control system configuration.

There exist several reliable control schemes [71, 76, 18, 35, 68, 37] that provide sta-

bility in the presence of a set of failed actuators and sensors that are non responsive.

However, in this chapter we have designed control schemes that guarantee stability in

the presence of malfunctioning actuators which continuously offer detrimental input

and thereby can lead the system to instability.

5.5 Summary

In this chapter, we designed two reliable control schemes using a local output

feedback control system that maintain asymptotic stability in the presence of Byzan-

tine actuators that continuously generate erroneous control inputs. The first scheme

was designed using redundant actuators that were colocated. However, it may not

be feasible to collocate actuators in all systems. The second scheme does not require

the actuators to be colocated. The other advantage with the second scheme is that

the required redundancy is reduced. ut in this scheme the restrictions in the choice of

actuator locations increased. The design of the system becomes more complex when

the number of state dimensions of the system increases. (Upper bounds for tolerating

186

Byzantine faults in higher dimension systems can be found in a related technical re-

port [47]). We gave an application of both the control schemes in stabilizing a beam

subjected to an initial perturbation.

We plan to extend our results on tolerating faulty actuators to systems that use

centralized and decentralized state feedback. An interesting topic for future study

is also to design reliable control schemes based on adaptive control laws using state

feedback. Extending some heuristic studies in this area [42] to sufficient conditions is

a subject of ongoing work.

187

CHAPTER 6

CONCLUDING REMARKS

6.1 Contributions

This dissertation has addressed the challenge of designing reliable applications

(that continue to grow in scale and complexity) using wireless sensor networks (that

continue to be resource constrained and unreliable in many ways). In order to address

this challenge, we have adopted a joint co-design of application and network layers,

and we propose the design and implementation of a network abstraction layer that

bridges the gap between the application and the network and provides performance

guarantees to enable design of robust applications. our approach is to identify and

specify the requirements of the application from the network in terms of suitable

network abstractions and then to implement these abstractions. The implementation

of the abstractions is performed by either using middleware services that are programs

running in the network or by simply designing the network by choosing the right

density or placement of nodes.

188

In this dissertation, we have designed reliable applications for classification, dis-

tributed tracking and distributed control using wireless sensor networks. In the con-

text of each of these applications, this dissertation has made specific contributions in

terms of application and network design. We list these below.

6.1.1 Snapshot services for wireless sensor networks

We have designed generalized snapshot services for arbitrary sensor networks with

N nodes that reside in an f -dimensional space, where each node periodically generates

m bits of information. Global state snapshots are a fundamental primitive for wireless

networks that sense and control real environments. Applications often require that

they be consistent and timely, which makes them potentially costly. Cost reduction is

often realized by gathering only a “delta” from previous snapshots. We have explored

an alternative form of efficiency by generalizing the notion of a snapshot to satisfy

distance sensitivity properties, wherein the state of nearby nodes is available with

greater resolution, speed, and frequency than that of farther away nodes.

Our algorithms can be formulated to allow delivery at a subset of nodes as opposed

to all nodes. They are memory efficient and realizable in networks with irregular

density, with arbitrary sized holes, imperfect clustering, and non unit disk radios. We

have quantified the maximum rate at which information can be generated at each node

so that snapshots are periodically delivered across the network. We have specified the

allowable aggregation functions in abstract terms, allowable functions include average,

max, min and wavelet functions. For our services, global time synchronization is

not required; a local notion of time however is needed to ensure fair scheduling of

transmission of nodes.

189

We have shown how our snapshot service can be used in a distributed pursuer

evader tracking application, where one or more pursuers are required to eventually

catch all the evaders in a region.

6.1.2 Trail: Sensor network service for distributed object

tracking

We have presented a wireless sensor network service, Trail, that supports distance

sensitive tracking of mobile objects for in-network subscribers upon demand. Trail

achieves a find time that is linear in the distance from a subscriber to an object,

via a distributed data structure that is updated only locally when the object moves.

Notably, Trail does not partition the network into a hierarchy of clusters and cluster-

heads, and as a result Trail has lower maintenance costs, is more locally fault-tolerant

and it better utilizes the network in terms of load balancing and minimizing the size

of the data structure needed for tracking. Moreover, Trail is reliable, and energy-

efficient, despite the network dynamics that are typical of wireless sensor networks.

Trail can be refined by tuning certain parameters, thereby yielding a family of pro-

tocols that are suited for different application settings such as rate of queries, rate of

updates and network size.

We have shown how Trail can be used to support a distributed pursuer evader

tracking application, commonly known as an asset protection game. We have evalu-

ated the performance of Trail by analysis, simulations in a 90-by-90 sensor network,

and experiments on 105 Mica2 nodes in the context of the pursuer-evader control

application.

190

6.1.3 Influence field based classification and tracking usingwireless sensor networks

We considered the problem of reliably estimating the influence fields of different

target types in a wireless sensor network subject to a variety of faults. We provided

mechanical procedures for sensor node density selection as well as algorithmic tech-

niques appropriate for dealing with each fault class. Corroboration of our results

and techniques was provided through at-scale experiments. We showed how reliable

estimation was achieved to enable accurate classification and tracking in A Line In

The Sand.

6.1.4 Reliable control system design despite Byzantine actu-

ators

Motivated by the application of wireless sensor networks in distributed control of

flexible structure, we designed two reliable control schemes using a local output feed-

back control system that maintain asymptotic stability in the presence of Byzantine

actuators that continuously generate erroneous control inputs. The first scheme was

designed using redundant actuators that were collocated. However, it may not be

feasible to collocate actuators in all systems. The second scheme does not require the

actuators to be collocated. The other advantage with the second scheme is that the

required redundancy is reduced. But in this scheme the restrictions in the choice of

actuator locations increased. The design of the system becomes more complex when

the number of state dimensions of the system increases. (Upper bounds for tolerating

Byzantine faults in higher dimension systems can be found in a related Technical re-

port [47]). We gave an application of both the control schemes in stabilizing a beam

subjected to an initial perturbation.

191

6.2 Future Work

This dissertation has identified the need to jointly design the application and net-

work layers so that (1)applications are designed in such a way that they are feasible to

implement given the network constraints (2) applications can be adaptive to varying

network conditions (3) the network can operate in such a way that the application

requirements are met and yet the system is optimal in its energy usage. By providing

network abstractions, applications can be designed without directly coupling them

with the network and therefore without exposing low level parameters such as those

of the MAC layer, thereby simplifying the application design. Some of the future

extensions of this work are as follows.

Adaptive Systems: In this dissertation, we have mostly looked at static configura-

tions where given a specification, network abstractions are implemented. Parameter-

izations for different application settings are done apriori. We would like to consider

adaptive system design, where the application and network layers are able to tune

dynamically to optimize overall system efficiency. This involves the application layer

and the middleware services to simultaneously tune themselves and dynamically agree

upon parameters.

Mobile wireless networks: While this dissertation has considered applications of

sensor networks where the network is static, of late sensors and actuators are inte-

grated into mobile objects such as cell-phones, PDAs and even humans and animals.

Mobility adds to the challenge of reliable application design and poses interesting

problems in the design of middleware services. Examples include location services

that are not dependent on static anchors and reliable transport services that can tol-

erate temporary disconnections and slow changing topologies. The location service

192

is aimed at providing low cost solutions for people/object tracking in indoor envi-

ronments and also to serve as a localization framework for other middleware services

such as routing that can run on top of it. For groups working in disaster relief situ-

ations or in remote and inaccessible regions such as forest workers and underground

miners, the location service and the reliable transport service will enable sharing

basic communication when other communication infrastructure is destroyed or not

available.

Control systems: This dissertation has also focused on control applications using

wireless sensor networks such as distributed vibration control. WSNs are ideally

suited for control of distributed parameter systems because of the scale at which

they can operate. Wireless networks also remove the need for cumbersome wiring

in industrial and process control applications. But when wireless networks are used

for control, reliability, low latency and security will be extremely critical. There

exists plenty of related work on network based control, that analyze the stability and

performance of control systems in the presence of latency and data losses. These

studies have focused on high bandwidth networks. How can these results be ported

to wireless networks? More importantly, what applications are feasible using wireless

networks? Can control strategies be relaxed to accommodate wireless networks?

These are interesting questions related to control systems using wireless networks.

High level specification: One of the contributions in this dissertation has been the

simplification of application design. We have achieved it in our work by abstracting

away the network details. The idea is that the application designers for wireless sensor

networks need not be networking experts. They should not be involved in the design

and implementation of the network. A future work in this direction is a high level

193

specification language for the application that results in translation to network level

specification and dynamically connecting to the appropriate abstraction.

Heterogeneity: As wireless sensor networks move into the urban domain, applica-

tions have to be developed across varied sensor networks. We characterize the urban

sensing problem as one where, in response to a specific requirement (such as tracking

and subsequent capture of an assailant), an application needs to be developed where

(a) the operations of varied, independent sensor networks are integrated and (b) the

information from these networks suitably retrieved, analyzed and fused to meet the

application. This points to not only a network abstraction oriented approach for

system design, but also standardization of these abstractions. As initial work in this

direction, we have recently prosed an architecture for composing applications across

heterogeneous sensor networks [48].

194

APPENDIX A

PROOF OF MAXIMA OF EXPRESSION IN EQ. 3.11

Proposition A.0.1. Let f(θ, φ) be defined as in Eq. A.1.

f(θ, φ) =(sin(θ) + sin(φ))

sin(θ + φ)(A.1)

The maximum value of f(θ, φ) where θ > 0, φ > 0, and 0 < (θ+φ) ≤ α is sec( α√2)

and occurs when θ = φ = α2.

Proof. To simplify, we use the following transformation matrix.

x =θ + φ√

2

y =−θ + φ√

2

where, 0 < x ≤ α√2

and |y| ≤ x.

Based on the above transformation, f(θ, φ) can be written in terms of x, y as

follows:

g(x, y) =(sin(x+y√

2) + sin(x−y√

2))

sin(√

2 ∗ x)(A.2)

195

Figure A.1: Finding maxima for f(θ, φ)

Now we first find the value of y at which function g(x, y) is maximum given a

value of x. For this we differentiate g(x, y) partially with respect to y and equate the

result to 0.

∂g(x, y)

∂y= 0

⇒(cos(x+y√

2) − cos(x−y√

2))

√2 ∗ sin(

√2 ∗ x)

= 0

⇒ y = 0

Thus we find that for function g(x, y) at any given value of x, y = 0 is the only

stationary point since x > 0. Differentiating g(x, y) partially twice with respect to y,

we note that result is less than 0 when y = 0.

∂2g(x, y)

∂y2

∣

∣

∣

y=0= − 1

cos( x√2)

< 0

Thus for any given value of x, g(x, y) is maximum when y = 0.

Also, when y = 0, g(x, y) = 1cos( x

√

2)which increases monotonically when x > 0 and

since x ≤ α√2, the maximum value is sec( α√

2) and the maximum occurs when x = α√

2,

i.e, (θ + φ) = α.

196

APPENDIX B

TECHNICAL NOTE: ON TERMINATING POINTS FORTRACKING MOBILE OBJECTS

In this section, we define the notion of terminating points for schemes that track

mobile objects in a distance sensitive manner in terms of update and find. We also

analyze the tradeoffs with respect to the choice of terminating points.

Let O be a set of mobile objects in a network of size N × N . Mobile objects are

of two types: finder objects (Of) and mover objects (Om). Thus O is a union of

disjoint sets Of and Om. Let p denote the location of any object P . Let Tracker

be a distance sensitive tracking scheme. Tracker maintains a track trackP for every

object P that belongs to Om. trackP is a set of points that contain information

pertaining to P . This information could be the actual state of P or simply a pointer

following which leads to the actual state of P . The length of trackP is the length

of the shortest curve connecting all points in trackP . Tracker offers two functions:

find(P, Q), that returns state of P to Q, where Q belongs to Of and P belongs to Op;

and move(P, p′, p) that updates trackP when P moves from p′ to p. Tracker satisfies

property F (find distance sensitivity) and property U (update distance sensitivity)

stated below:

197

Definition 17 (find distance sensitivity). Tracker satisfies property F if the cost of

find(P, Q) grows linearly with dist(p, q).

Definition 18 (update distance sensitivity). Tracker satisfies property U if the cost

of move(P, p′, p) grows linearly with dist(p, p′).

Note that in this document we consider only discrete moves of the objects. When

the motion of the object is continuous, a subset of Tracker schemes may have a

property that the cost of move is proportional to the distance of move in an amortized

sense.

From here on we assume that there is only one mover in the network and track

for that object is represented as track and we drop the identity subscript. We refer

to track for the mover at location x in the network simply as track for point x.

Definition 19 (Terminating Set τ). A terminating set τ in Tracker is a smallest set

of points such that track for every point in the network passes through at least one

point in τ .

The cardinality of a terminating set τ is denoted as µτ . The points contained in

a terminating set are called as terminating points of that set. Note that there could

also be multiple terminating sets in the network with each set containing an equal

but any number of points. In other words there are multiple smallest sets of points

such that track for every point in the network passes through at least one point in

each of those sets. For example in the case of Trail and Stalk [24], there is only

terminating set and this set has exactly one point, namely the center of the network.

Thus τ = C. In DSIB [27] and LLS [1], each set containing one publish location

at the highest level constitutes a terminating set. Thus in those schemes there are

198

multiple terminating sets but the cardinality of each set is 1. In the simple horizontal

vertical double ruling scheme, each horizontal line is a terminating set because every

vertical track passes through at least one point in each of those sets.

Lemma B.0.2. A terminating set τ cannot be an empty set.

Proof. track for each point in the network is at least equal to the point itself. Thus

τ cannot be empty.

In Trail, we have chosen a unique terminating set consisting of a unique termi-

nating point. We now analyze the tradeoffs involved when the terminating set is

not unique and when a terminating set contains more terminating points. We con-

sider 2 cases: a single terminating set with multiple terminating points and multiple

terminating sets each with one terminating point.

Case 1:µτ ≥ 1 We note that it is possible to decrease the maximum track length in

the network by dividing the network into regions and tracks being maintained with

respect to a terminating point in each region. (The disproportionate updates that

can be caused when an object keeps switching between boundaries can be avoided by

making the regions overlap.) The question then arises as to how small the regions can

be and yet maintain distance sensitivity. We now analyze the limits for decreasing

the track length and its effect on maximum find cost.

Given any location f of a finder, let Lf denote the length of the find trajectory

traversed from the finder location, after which the track for any point in the network

is found. Thus Lf denotes the worst case find cost from location f . Let Lf denote

the maximum value of Lf in the network.

Lemma B.0.3. For F to hold, Lf = O(N).

199

Proof. Note that the maximum distance between any two points in the network is

O(N). The result follows.

Lemma B.0.4. Lf is at least equal to the length of traversing all points in τ .

Proof. From the definition of a terminating set, τ is a minimum set of points through

which tracks from all points pass through.

Using Lemma A.6, we state the following Theorem.

Theorem B.0.5. The maximum find cost in the network is minimized when µτ = 1.

Proof. In the worst case, a find trajectory has to traverse all terminating points in the

terminating set. Compared to any configuration of terminating points when µτ > 1,

there exists a configuration of the terminating point when µτ = 1, such that Lf is

lower.

Using Lemmas A.5 and A.6, we get the following Lemma.

Lemma B.0.6. All terminating points in τ must be traversable in O(N).

Tha above Lemma imposes a lower bound the maximum size of the regions in the

following way. Let a region Rt be a set of points that choose t as a terminating point.

Let ρR denote the distance of the farthest point in the region from t. Let ρR denote

the maximum distance from a terminating point in any region of the network. We

state the following Theorem.

Theorem B.0.7. In order to preserve F , ρR = Ω(N).

Proof. Recall that all terminating points in τ must be traversed in O(N). If ρR <

Ω(N), then the maximum diameter of any region in the network is less than Ω(N) in

200

the N × N network and therefore all terminating points in τ cannot be traversed in

O(N).

Summary: The maximum track length in the network depends on the size of the

largest region. We note the maximum track length can be decreased by dividing

the network into regions and having local terminating points per region. However,

the size of the largest region can only be a constant order less than the diameter of

the network. For instance dividing the network into infinitesimally small regions or

even regions of size O(√

N) will violate F . Also when µτ > 1, find has to traverse

all the points in τ in the worst case. Thus the maximum find cost in the network

increases.

Case 2: Multiple terminating sets: Now we consider the case where there are

multiple terminating sets each with one terminating point and analyze the tradeoffs

in find and update cost. If there are multiple terminating sets then it is sufficient

for find to traverse the terminating point in any such set. In this case there is a

likelihood of decreasing the maximum find cost when compared to having only set

of terminating points because tracks can be found by reaching a terminating point

in any of the terminating sets. An example of this is the case where all tracks pass

through a common set of points as opposed to just one common point. Thus the

cardinality of each terminating point set is 1 because all tracks pass through each

point, but there are multiple such points.

Similar to case 1, we can show that the maximum find cost can be decreased by

dividing the network into regions and having one terminating set per region. We state

the following Theorems whose proofs are similar to that of case 1.

201

Theorem B.0.8. The maximum update cost in the network is minimized when the

number of terminating sets is 1.

Theorem B.0.9. In order to preserve U , ρR = Ω(N), where ρR is the maximum

distance of any point in the network from the terminating point in its local terminating

set.

Summary: The maximum find cost in the network depends on the size of the

largest region. We note the maximum find length can be decreased by dividing the

network into regions and having local terminating points per region. However, the

size of the largest region can only be a constant order less than the diameter of the

network. For instance dividing the network into infinitesimally small regions or even

regions of size O(√

N) will violate F . Also when the number of terminating sets is

greater than 1, update has to traverse the terminating point in all terminating sets

in the worst case. Thus the maximum update cost in the network increases.

Choice of unique terminating point: Maintaining a track with respect to

local terminating points and a single terminating set could be advantageous if it is

more likely that querying object and the object being found are closer and therefore

it is unlikely that all terminating points in the set have to be traversed. Similarly

maintaining a track with respect to multiple terminating sets could be advantageous

if objects are likely to move within bounded regions within a network. In this paper

we consider all distances between querying object and tracked object to be equally

likely and do not restrict mobility of the objects. Hence we consider only the case

where there is a unique terminating point, namely C.

Note that even if the network is divided into a constant number of regions, the

concept of maintaining a track with respect to any terminating point is the same

202

as in Trail. Note also that the tracks that are maintained in Trail with respect to

the terminating point are tight with a stretch factor that is less than 1.2 times the

shortest distance to the terminating point.

Multiple terminating sets with multiple terminating points: The cases

that we have presented can be extended to the case of multiple terminating sets, each

with multiple terminating points when there is more knowledge of find and update

patterns within each region.

203

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